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Millimeter Gravity and the Superstring Wall

by John G. Cramer

Alternate View Column AV-98
Keywords: superstring theory, gravity, Klein Kalusa, extra dimensions, loops, millimeter scale, inverse square law tests
Published in the December-1999 issue of Analog Science Fiction & Fact Magazine;
This column was written and submitted 06/08/99 and is copyrighted ©1999 by John G. Cramer.
All rights reserved. No part may be reproduced in any form without
the explicit permission of the author.


    Why is gravity so weak? Why are the color forces between quarks so strong? In the standard model of particle physics, why are there so many different energies at which distinct fundamental forces are supposed to "unify", and what determines these widely separated energies? The answers to these questions may be provided by extra dimensions curled into loops a millimeter around. In other words, our universe may be only a millimeter across, in directions we are not yet able to perceive. In this column we'll consider millimeter-size extra-dimensional loops and their implications.

    The idea of extra dimensions in physics can be traced back to 1919, a time when Einstein's general relativity was just emerging as the standard theory of gravity. Theodor Kalusa, a German mathematician then teaching at the University of Königsberg, sent a paper to Albert Einstein. In it he demonstrated how, by adding a 5th dimension to general relativity, Maxwell's formulation of electromagnetism could be incorporated into the theoretical framework of general relativity, a bold first step in formulating a "unified-field theory". Einstein provided encouragement, and Kalusa was able to publish his work in 1921. However, Einstein questioned the existence of Kalusa's extra dimension because, aside from the elegance of the mathematics, there was no evidence of such a dimension. He reasoned that if we really live in 5-dimensional space-time, we should have noticed this already.

    The puzzle of the missing dimension was partially resolved in 1926 when Oskar Klein, a colleague of Niels Bohr in Copenhagen, pointed out that Kalusa's extra dimension might be "hidden", provided it was "compactified" by being rolled up into a tiny loop of small size. Kalusa's extra-dimensional extension of general relativity combined with Klein's idea compactification of extra dimensions is now known as Kalusa-Klein (or K-K) theory, and after many decades of languishing on the sidelines, it has become a major theme of mainstream theoretical particle physics.

    A particle that has momentum directed along Kalusa's extra dimension will travel around the Klein loop and will soon arrive back where it started. Therefore, the particle is essentially confined to a "box" of the loop dimension. This causes momentum quantization it the loop direction because only certain wavelengths can fit around the loop. What we call electric charge, according to K-K theory, is actually motion in this dimension. Thus, charge is quantized by the loop condition.

    A charged particle must be travelling repeatedly around the K-K loop, even if it is at rest in normal space. If it moves, say, clockwise around the loop then it has a positive charge, while counterclockwise motion would give it a negative charge. The size of the unit charge and the strength of the electric force are inversely proportional to the distance around the loop: the smaller the loop, the larger is a unit charge. Newton's 3rd law of motion (action = reaction) applied to momentum along the K-K dimension leads to the law of conservation of electric charge, and the CPT theorem connecting the reversal of electrical charge with reversals in space and time directions is given a simple geometrical interpretation. K-K theory is "powerful" because it reveals connections between seemingly unrelated physical laws.

    The present "standard model" of fundamental particles (quantum chromodynamics or QCD) works too well to be "broken" by experiment, yet it is known to be a paste-up theory constructed around two dozen arbitrary variables: quarks and lepton masses, force strengths, and cross-connections between particles. All of these arbitrary variables must be put in "by hand", with no hint of their origin.

    And the Standard Model is daunting in another way. It predicts that as we go to higher particle energies with new accelerators, we will encounter an "energy desert", a vast wasteland where there are no new particles and not much interesting happens until energies at the Planck scale (1028 eV) are reached. Since we have no prospect of coming within many orders of magnitude of such a high energy, this is a very discouraging prediction for experimental particle physics.

    The Kalusa-Klein theory has become a key element of supersymmetric string theory, the best theoretical hope on the horizon for improving on this situation. String theory has an elegant mathematical structure that joins quantum mechanics and gravitation. It discards the usual view of fundamental particles as point size objects embedded in a three dimensional space with an additional time dimension. Instead, string theory describes fundamental particles as extended "string-like" objects embedded in 10 or more dimensions. The extra dimensions beyond 3-space and time, based on K-K theory, are compactified into loops perhaps at the Planck scale, about 10-35 m around, and are responsible for the fundamental strong, weak, and electromagnetic forces.

    For the past decade string theory has been the consuming passion of some of the best minds in theoretical physics, and enormous progress has been made. It has been realized that the many prescriptions for formulating string theory are limiting cases of the same theory (sometime called M-theory) and that multidimensional wall-like structures (called D-branes) can be found in the theory that simplify many of its problems.

    However, string theory still suffers from one basic difficulty: it cannot make experimentally testable predictions. The usual feedback loop linking hypothesis with experimental test, the dialogue that is the mainspring of the scientific method, is not applicable to string theory. In its present form, it can rely only on mathematical elegance to keep it on track. This is a very precarious situation for a developing theory. But perhaps this situation is about to change.

    At the beginning of this decade, a few string theorists began question the size of the extra-dimensional loops of string theory and to ask how big they might be before the theory came into conflict with observations. The answer, which is still being refined, is a surprise. Such loops can be as big as a millimeter, provided they involve the extra dimensions that are characteristic of the gravitational force. And millimeter-size extra dimensional loops for gravitation provides an explanation of why the force of gravity is so small compared to the other forces.

    The picture that emerges from these investigations is that each 4-space location in our universe is confined to a thin extra-dimensional "D-brane wall" on which the strong, weak, and electromagnetic interactions are confined, while the gravitational interaction is allowed to expand away from this wall in two or more extra dimensions which loop back on themselves in a distance of about a millimeter. Gravity, in this picture, is so weak because most lines of gravitational force are dissipated into these extra dimensions, while the other forces, confined to the wall, are not similarly weakened.

    There are many experimental implications of this variant of superstring theory. One is that energy scale of unification of the strong and electro-weak forces with gravitation is much lower that the standard model would predict, and may be reachable when the next large particle accelerator, the Large Hadron Collider or LHC, comes into operation at the CERN Laboratory in Geneva, Switzerland in 2005.

    But perhaps the more interesting prediction, because it will be tested sooner, is that at separation distances on the order of a millimeter, the gravitational force between two masses should become significantly stronger. This is because, if gravitational field lines are spreading out into extra dimensions compactified in millimeter loops, then when two gravitating objects are separated by distances of less than a millimeter, the extra field lines should appear. Gravity, instead of falling off like 1/r2 should begin to show a falloff of perhaps 1/r4. If r is a small number, dividing by it twice more makes the corresponding gravitational force much stronger.

    Unfortunately, this effect is not easy to observe in laboratory experiments. A typical "tabletop" measurement of the gravitational force uses a Cavendish balance, a pair of lead spheres about the size of tennis balls on a crossbar supported by a thin fiber. When similar spheres are placed nearby, the force of gravity twists the fiber, providing a measurement of the gravitational force. The centers of mass in such an experiment are separated by about 10 centimeters, so the distance scale at which gravity is being tested is about 100 times larger than that at which the interesting effects should appear. If one tries to simply make a Cavendish balance 100 times smaller, the gravitational forces, which are already small and scale with the volumes of the lead spheres, become (100)3=106 times smaller, much smaller than noise effects present in the experiment.

    However, an experiment designed by John C. Price and his co-workers at the University of Colorado promises to measure millimeter-scale gravitational forces. It will use a vibrating blade oscillating close to a "gravitational pickup" square that is isolated inside a shielding sapphire box and is part of another system that oscillates at the same frequency. The experimenters calculate that the varying gravitational force transmitted through the box should drive the oscillations of the pickup system and permit a determination of the gravitational force strength at millimeter separations. Other groups specializing in gravitational physics experiments are also designing experiments that will probe millimeter-scale gravitation. We can anticipate that in a year or so, well before the LHC at CERN goes into operation, we may have evidence for or against millimeter-scale superstring theories.

    What are the science fiction aspects of a universe with millimeter-scale compactified dimensions? The plot line of my hard SF novel Twistor (Morrow, 1989) used a "shadow Earth", a planet-twin of the Earth occupying the same orbit around a similar sun, but made of "shadow matter", a projection of matter in an extra K-K dimension which interacts with the matter of our planet only gravitationally. Although my novel was written before the theoretical developments described above, superstring theory with millimeter-scale dimensions provides an additional theoretical basis for such an extrapolation.

    Our universe is puzzling because certain expected particles have not been found. In particular, missing from our universe in the expected quantities are: antimatter particles (antiprotons and positrons), magnetic monopoles, and six right-handed weakly interacting particles (Z, W, and three flavors of neutrinos). Further, we know that 5/6 or more of the non-vacuum mass in our universe is in the form of "dark matter" that shows up only through its gravitational interactions. It is fair to ask where the dark matter "lives" and where the missing particles have gone.

    The superstring theory discussed above describes our universe as having the strong, weak, and electromagnetic interactions confined to a compactified "wall" while the gravitational interaction is allowed to spread out over a millimeter of distance into two or more extra K-K dimensions. A millimeter is a huge span of distance when compared to the 10-31 meter size of the other compactified loops. What else might lie in that millimeter? Why should there be only one such wall there?

    Perhaps there are other walls wherein the missing dark matter, antimatter, magnetic monopoles, right-handed weak particles, and particles we haven't even dreamed of may reside. Perhaps there are Twistor-like shadow worlds made of normal or exotic matter, "parallel" to ours but lying on another wall, separated from ours by only a fraction of a millimeter, but completely unconnected to our world except through the effects of gravity.

    Perhaps we are on the threshold of discovering these extra dimensions and our "wall" embedded in them though their gravitational and particle-physics effects. Perhaps we stand on the threshold of a doorway that leads to extra dimensions. Watch this column for further developments.


Millimeter loops in superstring theory:

"Phenomenology, Astrophysics, and Cosmology of Theories with Sub-Millimeter Dimensions and TeV Scale Quantum Gravity", Nima Arkani-Hamed, Savas Dimopoulos, and Gia Dvali, Physical Review D59 (1999) 860-904, preprint hep-ph/9807344 in the LANL Archive, (July 12, 1998).

Millimeter loops in superstring theory:

"Experimental Status of Gravitational-Strength Forces in the Sub-Centimeter Regime", J. C. Long, H. W. Chan, and J. C. Price, Nuclear Physics B539 (1999) 23-3, preprint hep-ph/9805217 in the LANL Archive, (April 2, 1999).

SF Novels by John Cramer:  my two hard SF novels, Twistor and Einstein's Bridge, are newly released as eBooks by Book View Cafe and are available at : .

AV Columns Online: Electronic reprints of about 177 "The Alternate View" columns by John G. Cramer, previously published in Analog , are available online at:

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