How many dimensions are there in space?
In 1884 Edwin A. Abbott described in his bizarre novel "Flatland: A Romance of Many Dimensions" the adventures of the square being "A. Square" in a two-dimensional flat world populated by geometric creatures - triangles, squares, pentagons and so on is. Towards the end of the story, on the first day of the year 2000, a spherical creature from the three-dimensional "Spaceland" crosses the flat land and lifts A. Square out of its flat living space in order to show him the three-dimensional big wide world. This gives A. Square the idea that the space land might again be just a small subspace of an even larger four-dimensional universe.
In fact, in recent years some physicists have begun to seriously pursue a very similar idea: that everything we perceive in our universe is limited to a three-dimensional "membrane" that lies in a higher-dimensional area. But unlike A. Square, who only came to his insights with the help of higher interference from space land, the physicists may soon find out the existence of additional dimensions for themselves. Attempts have already been made to demonstrate the effects of these extra dimensions on gravity. And if the theory is correct, experiments planned in the near future in high-energy particle accelerators could reveal exotic processes at the limit of quantum and gravity theory - for example the short-term generation of tiny black holes. This theory is more than just "romance in many dimensions"; it is based on the latest developments in string theory and will potentially solve some persistent puzzles in cosmology and particle physics.
The inexplicable weakness of gravity
Terms like strings and extra dimensions actually originate from an attempt to understand the most familiar of all natural forces: gravity. More than three centuries after Isaac Newton formulated his law of gravitation, physics still cannot explain why gravity is so much weaker than all other interactions. A small magnet, when it lifts a nail from the ground, easily overcomes the pull of the entire mass of the earth. The gravitational attraction between two electrons is 10E43 times weaker than the electrical repulsion between them. Gravity presses our feet on the ground and makes the earth revolve around the sun - but only because these enormous masses are electrically neutral. That is why the electrical forces remain negligible, and gravity is the only one noticeable in spite of its weakness.
The electron mass would have to be 10E22 times as large as its actual value so that gravity and electrical interaction would be equally strong. To produce such a heavy particle, an energy of 10E19 gigaelectron volts (GeV, billions of electron volts) would be required - the so-called Planck energy. The Planck length, which is only a tiny 10E-35 meters, is related to it. For comparison: the nucleus of the hydrogen atom, the proton, is about 10E19 times as large and has a mass of about 1 GeV. Planck energy and Planck length - together called the Planck scale - are far beyond the range of the most powerful particle accelerators. Even the Large Hadron Collider at Cern, when it goes into operation in five years' time, will only be able to examine lengths of at least 10E-19 meters (see "The Great Hadron Collider" by Chris Llewellyn Smith, Spektrum der Wissenschaft 9/2000, P. 68). Since in the area of the Planck scale the gravitation becomes as strong as the electromagnetism and the other natural forces, the physicists assumed that only with such enormous energies a "theory for everything" would reveal itself, the great union of the gravitation with the other forces. Accordingly, the unified theory would be hopelessly beyond the reach of direct experimental verification in the foreseeable future (see "A Theory for Everything?" By Steven Weinberg, Spectrum Special 1/2000 "Research in the 21st Century", p. 12).
The most powerful accelerators today achieve energies between 100 GeV and 1 TeV (teraelectron volts, trillion electron volts). In this area, electromagnetism is combined with the so-called weak interaction, a force between subatomic particles responsible for certain radioactive forms of decay. We would understand the extraordinary weakness of gravity if we could explain the huge factor 10E16 that separates the electroweak scale from the Planck scale.
Unfortunately, the extremely successful standard model of particle physics is not sufficient for this, because the model was specially adapted to the experimentally observed value of the electroweak scale. The good news is that this adjustment - along with 16 others - is enough to explain hundreds of thousands of observations in one fell swoop. The bad news is that we have to set the underlying theory to the thirty-second decimal place; otherwise the electroweak scale would assume the extreme values of the Planck scale due to quantum physical instabilities. It's like walking into a room and finding a pencil that is perfectly balanced on its tip in the middle of the table. While not impossible, such a situation is extremely unstable and one wonders where it is coming from.
"Large" room dimensions
For twenty years theorists have been trying to solve this riddle - the hierarchy problem - by changing the particle physics at around 10E-19 meters (or 1 TeV) in order to stabilize the electroweak scale. The most common change to the standard model achieves this purpose via what is known as supersymmetry. To stay in the picture of the balanced pencil: The supersymmetry works like an invisible thread that pulls the pencil upwards and prevents it from tipping over. Although the particle accelerators have not yet found any direct indications of supersymmetry, there are at least indirect indications. If, for example, the measured strengths of the strong, weak and electromagnetic interaction are theoretically extrapolated to ever shorter distances, then they only converge very precisely in a common value if the extrapolation obeys the rules of supersymmetry. This result indicates a supersymmetric union of the three forces at about 10E-32 meters; that is around a thousand times greater than the Planck length, but still far beyond the range accessible to particle accelerators.
However, in the past two years, some theorists have proposed a radically new approach that modifies spacetime, gravity, and the Planck scale itself. The basic idea is that the extreme values of the Planck scale - which have been accepted since the German physicist Max Planck (1858–1947) introduced them a century ago - are based on an untested assumption about gravity at short distances.
Newton's law of gravitation states that the force between two masses is inversely proportional to the square of their distance; it works great over macroscopic distances and explains the earth's orbit around the sun, the moon's orbit around the earth, and so on. But because gravity is so weak, the law was only checked experimentally up to a distance of around one millimeter - and at least we have to extrapolate over 32 orders of magnitude to conclude that gravity is only strong at a Planck scale of 10E-35 meters becomes.
The membrane model of the universe
The law of the inverse square of the distance results entirely in three-dimensional space. Let us consider the field lines of gravity, which emanate uniformly from the earth. At a greater distance from the earth, they are distributed over a correspondingly larger spherical surface: the surface increases with the square of the radius, and the force is diluted to the same extent. Suppose there was one more dimension, space was four-dimensional. Then the field lines emanating from a point would spread over a four-dimensional spherical shell, the surface of which would grow to the third power of the radius, and gravity would obey a law of the inverse third power of the distance.
This inverse cubic law certainly does not describe our universe, but let's imagine that the additional dimension is curled up into a small circle with radius R. Let us now consider field lines that start from an almost point-like mass. Over very small distances - much smaller than R - the field lines can spread out uniformly in all four dimensions, and therefore the force of gravity is inversely proportional to the cube of the distance. Once the field lines have expanded completely around the circle, they only have three dimensions left. Therefore, for distances that are much larger than R, the force is inversely square.
The same applies to any number of extra dimensions, all of which are rolled into circles with radius R. With n additional dimensions, the force of gravity for distances below R follows an inverse power law with the power 2 + n. Because we have only measured gravity for distances above a millimeter, we would not even notice changes in gravity due to extra dimensions whose size R is less than a millimeter. In addition, the (2 + n) -Potenzgesetz would have the effect that the force of gravity does not only become "strong" when the conventional Planck scale of 10E-35 meters is reached, but far above. That is, the Planck length - defined as the length at which gravity becomes strong - would not be so tiny, and the hierarchy problem would be less.
The hierarchy problem can even be completely solved by postulating so many extra dimensions that the Planck scale shifts into the vicinity of the electroweak order of magnitude. The final union of gravity with the other forces would then take place at 10E-19 meters and not - as previously assumed - at 10E-35 meters. How many dimensions you need depends on how big they are. Conversely, if we give a number of additional dimensions, we can calculate how big they need to be to make gravity strong at 10E-19 meters. With only one additional dimension, its radius R must be about as large as the distance between the earth and the sun. Therefore, this case is already excluded by observation. But even two extra dimensions can solve the hierarchy problem if they are around a millimeter in size - and it is precisely at this limit that our direct knowledge of gravity ends. The dimensions are even smaller if we take more of them: seven additional dimensions only need to be 10E-14 meters - the size of a uranium core. This is tiny in everyday terms, but still huge on the scale of particle physics.
Postulating additional dimensions may seem bizarre and arbitrary, but it's a familiar idea to theorists. As early as the 1920s, the physicists Theodor Kaluza (1885–1954) and Oskar Klein (1894–1977) developed a unified theory of gravitation and electromagnetism, which required an additional dimension. The idea returns in modern string theories, which for mathematical reasons require a total of 10 spatial dimensions. So far, physicists have assumed that the extra dimensions are rolled up into tiny circles the size of the conventional Planck length - 10E-35 meters - whereby they remain hidden, but leave the dilemma of the hierarchy problem. In contrast, in the new theory that we propose, the additional dimensions are rolled up into relatively large circles - at least 10E-14 meters, a maximum of one millimeter.
If these dimensions are so large, why haven't we noticed them before? Millimeter-sized extra dimensions could already be perceived with the naked eye and even more so through a microscope. And although we have not measured gravity below a millimeter, we have a wealth of experimental knowledge about all other forces at much shorter distances, down to 10E-19 meters - and all of this can only be reconciled with a three-dimensional space. Then how can there be great extra dimensions?
The answer is simple and strange at the same time: All matter and all forces known to us - with the exception of gravity - are limited to a kind of wall in the space of additional dimensions. Electrons, protons, photons and all the other particles of the Standard Model cannot move in the extra dimensions; electric and magnetic field lines do not spread into the higher-dimensional space either. The wall is only three-dimensional, and as far as these particles are concerned, the universe might as well be three-dimensional. Only the field lines of gravity reach out into the higher-dimensional space, and only the graviton - the quantum particle that transmits gravity - is able to move freely there. This means that the additional dimensions are only noticeable through gravity.
What is the theory good for?
To illustrate this, let's imagine all the particles in the Standard Model as balls on an immeasurably large pool table; as far as they are concerned, the universe is two-dimensional. Nevertheless, residents of this billiard universe can discover the higher-dimensional world: When two billiard balls collide, sound waves are generated that propagate in all three dimensions and allow a little energy to disappear from the table surface. The sound waves correspond to the gravitons, which can move in the entire higher-dimensional space. In the case of high-energy particle collisions, we should therefore observe certain energy deficits that result from gravitons that have escaped into higher dimensions. While it may seem strange to us that some particles should be confined to a wall, we are familiar with similar phenomena. For example, electrons in a copper wire can only move in the one-dimensional space of the wire; they do not migrate into the three-dimensional environment. Water waves also propagate on the surface, not in the depths. Our special scenario, in which all particles except gravity are confined to a wall, follows easily from string theory. In fact, the latest breakthrough in string theory is related to such walls or membranes, so-called "D-branes"; This made-up word is composed of "D" for Dirichlet - a German mathematician of the 19th century, after whom certain boundary conditions for fields are named - and "Brane" from English membrane. D-branes have exactly the required properties: electrons, photons and other particles are described by tiny vibrating strings, the two end points of which have to adhere to a D-brane. In contrast, the gravitons are tiny closed string loops; they can wander around in all dimensions because they have no endpoints that would be anchored in a D-brane.
A good researcher tries to get a new theory done right away by finding a contradiction to known experimental results. The theory of the large extra dimensions changes the force of gravity at macroscopic distances and the rest of the physics at high energies - so it should actually be easy to invalidate. But surprisingly, this theory, although it deviates radically from our usual picture of the universe, does not contradict any known experimental result. A few examples show how surprising this conclusion is.
One might initially expect that a change in gravity will affect the objects it holds together, such as stars and galaxies. But that is not the case. Gravity only changes for distances less than a millimeter while holding a star together for thousands of kilometers. Generally speaking: Although the gravitation is increased over short distances by the additional dimensions, it catches up with the other forces only at 10E-19 meters; at great distances it remains comparatively very weak.
A much more serious problem is the gravitons, the hypothetical quanta of gravitation. In our theory, because of the increased force of gravity over a short distance, they interact much more strongly with matter, and that is why many more gravitons should be generated in high-energy particle collisions. In addition, they reproduce in all dimensions and therefore abduct energy from the wall or membrane that makes up our universe.
If a star collapses and then explodes as a supernova, gravitons can easily evaporate into the extra dimensions at the high temperatures. But as we know from observations of the famous 1987A supernova, such an explosion emits most of its energy in the form of neutrinos; there is hardly any leeway for energy loss through gravitons. Our knowledge of supernovae therefore places a narrow limit on the interaction of gravitons with matter. This restriction almost brought the death blow to the idea of extra dimensions; but detailed calculations show that the theory survives.The narrowest bound applies to only two additional dimensions; in this case, gravitons cool the supernova too much if the fundamental Planck scale is lowered to less than around 50 TeV. With three or more extra dimensions, this size may even be only a few TeV without the supernova extinguishing prematurely.
In theory, many other systems - from the successful Big Bang model of the early universe to the collision of cosmic rays of the highest energy - have been investigated to see what limitations they impose on the new theory. The theory passes all of these experimental tests; they are even less strict than the supernova restriction. The more dimensions are added to the theory, the looser the restrictions become: The dramatic increase in gravity begins at smaller distances and therefore has fewer effects on large-scale processes.
Clarity in 2010
The theory solves the hierarchy problem by turning gravity into a strong force in the TeV energy range - especially in the range that the planned particle accelerators are supposed to investigate. According to this, the Large Hadron Collider (LHC), which will start work around 2005, could reveal the nature of quantum gravity. If string theory describes quantum gravity correctly, the particles resemble tiny string loops that can vibrate like a violin string. The well-known fundamental particles correspond to a string that does not vibrate - a string that is not bowed. Every different tone that the vibrating string is able to produce corresponds to a new exotic particle in this picture. According to conventional string theory, the strings should only be about 10E-35 meters in size, and the new particles would have masses on the order of magnitude of the conventional Planck energy. The music on these strings would be too shrill for us to hear it with our accelerators. But with large extra dimensions, the strings are much longer, namely around 10E-19 meters, and the new particles can already appear at a few TeV - deep enough to be heard with the LHC.
Likewise, energies could be experimentally attainable at which particle collisions produce microscopic black holes. With a diameter of around 10–19 meters, these structures would be too small to create problems; they would emit energy in the form of what is known as Hawking radiation and evaporate in less than 10E-27 seconds. By observing such phenomena, the enigmatic quantum physics of black holes can be directly explored.
Even at energies that are too low to produce oscillating strings or black holes, the particle collisions produce large quantities of gravitons - a process that is insignificant in conventional theories. Experimentally, the emitted gravitons are not directly detectable, but the energy carried away by them would reveal itself as an energy deficit of the collision debris. The theory predicts certain properties of the missing energy - such as how it varies with the collision energy. In this way, graviton generation can be distinguished from other processes that remove energy in the form of invisible particles. The data from the most powerful high-energy accelerators are already restricting the scenario of the large extra dimensions somewhat. The experiments at the LHC should either find evidence of gravitons or, if not, refute the theory.
A completely different type of experiment could also support the theory, perhaps even earlier than the particle accelerator. As we know, two extra dimensions have to be around a millimeter in size to solve the hierarchy problem. Then gravity measurements at millimeter distances would show a transition from Newton's inverse quadratic law to a law with the inverse fourth power of the distance. Extensions to the basic theory lead to numerous other possible deviations from Newton's law; Most interesting are repulsive forces, which are more than a million times stronger than gravity when two particles are less than a millimeter apart. Currently, extremely sensitive detectors that can be placed on a laboratory bench are checking Newton's law of gravity in the range of centimeters to a few hundredths of a millimeter.
In order to test the force of gravity below a millimeter distance, the objects must not be much larger than a millimeter; therefore they have only very small masses. Numerous effects have to be screened very carefully - such as residual electrostatic charges that mask or imitate the tiny gravitational attraction. Such experiments are difficult and subtle, but also tremendously exciting, because they could reveal completely new physics. Even regardless of the search for additional dimensions, it is important to extend our immediate knowledge of gravity to these short distances. Three researchers are currently conducting such experiments: John Price from the University of Colorado, Aharon Kapitulnik from Stanford University, and Eric G. Adelberger from the University of Washington. They expect preliminary results this year.
The idea of additional dimensions actually continues the tradition of our Copernican worldview: the earth is not the center of the solar system, the sun is not the center of our galaxy, our galaxy is just one of billions in a universe without a center - and now our whole is formed three-dimensional universe just a thin membrane in a space with many dimensions. If we look at sections through the extra dimensions, our universe only takes up a single, infinitely small point, surrounded by emptiness.
But maybe that's not the whole truth. Just as the Milky Way is not the only galaxy in the universe, our universe may not be alone in the extra dimensions. The membranes of other three-dimensional universes could lie parallel to ours, in the extra dimensions only a millimeter away from us. And although all particles of the Standard Model are trapped in our membrane universe, besides the gravitons, other particles that do not belong to the Standard Model could propagate through the extra dimensions. Far from being empty, the additional dimensions may have a variety of interesting structures.
The effects of new particles and universes in the extra dimensions may solve many puzzles in particle physics and cosmology. For example, you could be responsible for the neutrino mass. Impressive new results from the Super Kamiokande experiment in Japan indicate that the neutrinos, long thought to be massless, have a tiny mass (see "Tracking the neutrino mass" by Edward Kearns, Takaaki Kajita and Yoji Totsuka, Spectrum of Science 10/1999, p. 44). The neutrino could acquire its mass through interaction with a partner field in the extra dimensions. As with gravity, the interaction would be very dilute - and the neutrino mass tiny - because the partner spreads through the extra dimensions.
Another mystery of cosmology is the question of what dark matter is made of: the invisible substance, recognizable only by its gravitational attraction, which seems to make up more than 90 percent of the mass of the universe. Maybe she is in parallel universes. Such matter would affect our universe through gravity; it would necessarily be "dark" because our kind of photons are irrevocably trapped in our membrane, and therefore light can never penetrate the void that separates us from parallel matter.
Such parallel universes are perhaps completely different from ours: They consist of a membrane with fewer or more dimensions and contain completely different particles and forces. Or even stranger, they even have the same properties as our world. Suppose our home membrane is folded several times in the extra dimensions. Objects on an opposite crease then appear to be very far away, although less than a millimeter separates them from us in the extra dimensions: the light they emit must take the entire detour through the crease to us. If the fold is tens of billions of light years across, then no ray of light has reached us from the other side since the beginning of the universe.
The mysterious dark matter could consist of normal matter, perhaps even ordinary stars and galaxies that shine brightly on their side of the fold. Such stars would produce interesting observable effects - such as gravitational waves from supernovae and other violent astrophysical processes. The gravitational wave detectors, which should be completed in a few years, could find signs of wrinkles: large sources of gravitational radiation, to which no visible matter in our universe can be assigned.
The theory presented here is not the first with additional dimensions greater than 10E-35 meters. Ignatios Antoniadis of the École Polytechnique in France suggested in 1990 that some dimensions of string theory could be up to 10E-19 meters, but he left the scale of quantum gravity at 10E-35 meters. And in 1996 Petr Horava from the California Institute of Technology and Edward Witten from the Institute for Advanced Study in Princeton, New Jersey, pointed out that a single extra dimension, 10E-30 meters in size, could unite gravity with the other forces as part of a supersymmetric union at 10E-32 meters. Following this idea, Joseph Lykken from the Fermi National Accelerator Laboratory in Batavia (Illinois) tried to reduce the size of the association to 10E-19 meters - but without introducing large extra dimensions. As Keith Dienes from the University of Arizona and Emilian Dudas and Tony Gerghetta at Cern found out in 1998, extra dimensions that are smaller than 10E-19 meters would allow a union of all natural forces further above 10E-32 meters.
Our universe: just one among many?
Since our proposal in 1998, several interesting variants have appeared that also use extra dimensions and our membrane universe. Lisa Randall from Princeton University and Raman Sundrum from Stanford University assume that gravity itself is concentrated on a membrane in five-dimensional space-time that is infinite in all directions. Gravity naturally appears very weak in our universe when we are on another membrane.
For twenty years it was customary to explain the hierarchy problem and thus the weakness of gravity by assuming that the Planck scale at 10E-35 meters is the basis of every theory, and that particle physics must change at 10E-19 meters. Quantum gravity remained pure speculation and hopelessly beyond the reach of experimentation. Over the past two years, we've realized that this doesn't necessarily have to be the case. If there are large additional dimensions, in the next few years we could discover certain deviations from Newton's law at around 6 x 10E-5 meters, as well as string vibrations or tiny black holes with the help of the LHC. Quantum gravity and string theory would become part of the experimentally verifiable science. In any case, by 2010 we will get closer to the answer to the 300 year old question, why gravity is so weak. Perhaps we will then find ourselves in a strange flat land - in a membrane universe, where quantum gravity is within reach.
The elegant universe. Superstrings, hidden dimensions and the search for the world formula. Brian Greene Siedler, Berlin 2000.
New world theories: from strings to membranes. By M. Duff in: Spectrum of Science 4/1998, p. 62.
Flatland. A Romance of Many Dimensions. By Edwin A. Abbott. The text is available on the Internet at http://promo.net/cgi-promo/pg/t9.cgi?entry=97
Our universe obviously has four dimensions: three spatial and one temporal. But mathematicians and physicists have long been researching the properties of abstract spaces with any number of dimensions.
The "size" of dimensions
The four known space-time dimensions of our universe are huge. The dimension of time extends at least 13 billion years into the past and perhaps infinitely far into the future. The three dimensions of space are perhaps infinite; our telescopes capture objects more than 12 billion light years away. But dimensions can also be finite. For example, the two dimensions of the earth's surface only reach about 40,000 kilometers.
Small extra dimensions
Some modern physical theories posit additional real-world dimensions curled up into circles so tiny - perhaps only 10E-35 meters in radius - that we have not yet discovered them. A cotton thread is one-dimensional as a good approximation: A single number can indicate where an ant is sitting on the thread. But under the microscope we see dust mites crawling on the two-dimensional surface of the thread - along the large length dimension as well as the small circumferential dimension.
Big extra dimensions
Physicists recently realized that there could be millimeter-sized yet invisible extra dimensions. Surprisingly, this theory does not conflict with previously known experimental facts, and it could clear up some puzzles in particle physics and cosmology. According to this, our entire spatial universe - with the exception of gravity - would be trapped in a membrane like billiard balls on a two-dimensional gaming table.
Dimensions and gravity
The behavior of gravity - especially its strength - is closely related to the number of dimensions accessible to it. Gravitational measurements over distances of less than a millimeter - such experiments are currently in progress - could therefore reveal large extra dimensions. These dimensions would also bring hypothetical quantum gravity objects within reach.
From: Spectrum of Science 10/2000, page 44
© Spektrum der Wissenschaft Verlagsgesellschaft mbH
This article is included in Spectrum of Science 10/2000
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