The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory

by Brian Greene

  5. With the discovery of M-theory and the recognition of an eleventh dimension, string theorists have begun studying ways of curling up all seven extra dimensions in a manner that puts them all on more or less equal footing. The possible choices for such seven-dimensional manifolds are known as Joyce manifolds, after Domenic Joyce of Oxford University, who is credited with finding the first techniques for their mathematical construction.

  6. Interview with Cumrun Vafa, January 12, 1998.

  7. The expert reader will note that our description is taking place in the so-called string frame of reference, in which increasing curvature during the pre-big bang arises from (a dilaton-driven) increase in the strength of the gravitational force. In the so-called Einstein frame, the evolution would be described as an accelerating contraction phase.

  8. Interview with Gabriele Veneziano, May 19, 1998.

  9. Smolin's ideas are discussed in his book The Life of the Cosmos (New York: Oxford University Press, 1997).

  10. Within string theory, for example, this evolution could be driven by small changes to the shape of the curled-up dimensions from one universe to its offspring. From our results on space-tearing conifold transitions, we know that a sufficiently long sequence of such small changes can take us from one Calabi-Yau to any other, allowing the multiverse to sample the reproductive efficiency of all universes based on strings. After the multiverse has passed through sufficiently many stages of reproduction, Smolin's hypothesis would lead us to expect that the typical universe will have a Calabi-Yau component that is optimized for fertility.

  Chapter 15

  1. Interview with Edward Witten, March 4, 1998.

  2. Some theorists see a hint of this idea in the holographic principle, a concept originated by Susskind and the renowned Dutch physicist Gerard 't Hooft. Just as a hologram can reproduce a three-dimensional visual image from a specially designed two-dimensional film, Susskind and 't Hooft have suggested that all of the physical happenings we encounter may actually be encoded fully through equations defined in a lower-dimensional world. Although this may sound as strange as trying to draw someone's portrait by viewing only their shadow, we can get a sense of what it means, and understand part of Susskind's and 't Hooft's motivation, by thinking about black hole entropy as discussed in Chapter 13. Recall that the entropy of a black hole is determined by the surface area of its event horizon—and not by the total volume of space that the event horizon bounds. Therefore, the disorder of a black hole, and correspondingly the information it can embody, is encoded in the two-dimensional data of surface area. It is almost as if the event horizon of the black hole acts like a hologram by capturing all the information content of the black hole's three-dimensional interior. Susskind and 't Hooft have generalized this idea to the whole universe by suggesting that everything that occurs in the "interior" of the universe is merely a reflection of data and equations defined on a distant, bounding surface. Recently, work by the Harvard physicist Juan Maldacena, together with important subsequent work by Witten and of Princeton physicists Steven Gubser, Igor Klebanov, and Alexander Polyakov, has shown that, at least in certain cases, string theory embodies the holographic principle. In a manner that is currently being investigated vigorously, it appears that the physics of a universe governed by string theory has an equivalent description that involves only physics that takes place on such a bounding surface—a surface necessarily of lower dimensionality than the interior. Some string theorists have suggested that fully understanding the holographic principle and its role in string theory may well lead to the third superstring revolution.

  3. Sir Isaac Newton’s Mathematical Principles of Natural Philosophy and His System of the World, trans. Motte and Cajori (Berkeley: University of California Press, 1962), Vol. 1, p. 6.

  4. If you are familiar with linear algebra, one simple and relevant way of thinking about noncommutative geometry is to replace conventional Cartesian coordinates, which commute under multiplication, with matrices, which do not.

  5. Interview with Cumrun Vafa, January 12, 1998.

  6. Interview with Edward Witten, May 11, 1998.

  7. Quoted in Banesh Hoffman with Helen Dukas, Albert Einstein, Creator and Rebel (New York: Viking, 1972), p. 18.

  8. Martin J. Klein, "Einstein: The Life and Times, by R. W. Clark," (book review) Science 174, pp. 1315-16.

  9. Jacob Bronkowski, The Ascent of Man (Boston: Little, Brown, 1973), p. 20.

  * * *

  Glossary of Scientific Terms

  Absolute zero. The lowest possible temperature, about -273 degrees Celsius, or 0 on the Kelvin scale.

  Acceleration. A change in an object's speed or direction. See also velocity.

  Accelerator. See particle accelerator.

  Amplitude. The maximum height of a wave peak or the maximum depth of a wave trough.

  Anthropic principle. Doctrine that one explanation for why the universe has the properties we observe is that, were the properties different, it is likely that life would not form and therefore we would not be here to observe the changes.

  Antimatter. Matter that has the same gravitational properties as ordinary matter, but that has an opposite electric charge as well as opposite nuclear force charges.

  Antiparticle. A particle of antimatter.

  ATB. Acronym for "after the bang"; usually used in reference to time elapsed since the big bang.

  Atom. Fundamental building block of matter, consisting of a nucleus (comprising protons and neutrons) and an orbiting swarm of electrons.

  Big bang. Currently accepted theory that the expanding universe began some 15 billion years ago from a state of enormous energy, density, and compression.

  Big crunch. One hypothesized future for the universe in which the current expansion stops, reverses, and results in all space and all matter collapsing together; a reversal of the big bang.

  Black hole. An object whose immense gravitational field entraps anything, even light, that gets too close (closer than the black hole's event horizon).

  Black-hole entropy. The entropy embodied within a black hole.

  Boson. A particle, or pattern of string vibration, with a whole number amount of spin; typically a messenger particle.

  Bosonic string theory. First known string theory; contains vibrational patterns that are all bosons.

  BPS states. Configurations in a supersymmetric theory whose properties can be determined exactly by arguments rooted in symmetry.

  Brane. Any of the extended objects that arise in string theory. A one-brane is a string, a two-brane is a membrane, a three-brane has three extended dimensions, etc, More generally, a p-brane has p spatial dimensions.

  Calabi-Yau space, Calabi-Yau shape. A space (shape) into which the extra spatial dimensions required by string theory can be curled up, consistent with the equations of the theory

  Charge. See force charge.

  Chiral, Chirality. Feature of fundamental particle physics that distinguishes left- from right-handed, showing that the universe is not fully left-right symmetric.

  Closed string. A type of string that is in the shape of a loop.

  Conifold transition. Evolution of the Calabi-Yau portion of space in which its fabric rips and repairs itself, yet with mild and acceptable physical consequences in the context of string theory. The tears involved are more severe than those in a flop transition.

  Cosmic microwave background radiation. Microwave radiation suffusing the universe, produced during the big bang and subsequently thinned and cooled as the universe expanded.

  Cosmological constant. A modification of general relativity's original equations, allowing for a static universe; interpretable as a constant energy density of the vacuum.

  Coupling constant. See string coupling constant.

  Curled-up dimension. A spatial dimension that does not have an observably large spatial extent; a spatial dimension that is crumpled, wrapped, or curled up into a tiny size, thereby evading direct detection.

  Curvature. The
deviation of an object or of space or of spacetime from a flat form and therefore from the rules of geometry codified by Euclid.

  Dimension. An independent axis or direction in space or spacetime. The familiar space around us has three dimensions (left-right, back-forth, up-down) and the familiar spacetime has four (the previous three axes plus the past-future axis). Superstring theory requires the universe to have additional spatial dimensions.

  Dual, Duality, Duality symmetries. Situation in which two or more theories appear to be completely different, yet actually give rise to identical physical consequences.

  Electromagnetic field. Force field of the electromagnetic force, consisting of electric and magnetic lines of force at each point in space.

  Electromagnetic force. One of the four fundamental forces, a union of the electric and magnetic forces.

  Electromagnetic gauge symmetry. Gauge symmetry underlying quantum electrodynamics.

  Electromagnetic radiation. The energy carried by an electromagnetic wave.

  Electromagnetic wave. A wavelike disturbance in an electromagnetic field; all such waves travel at the speed of light. Visible light, X rays, microwaves, and infrared radiation are examples.

  Electron. Negatively charged particle, typically found orbiting the nucleus of an atom.

  Electroweak theory. Relativistic quantum field theory describing the weak force and the electromagnetic force in one unified framework.

  Eleven-dimensional supergravity. Promising higher-dimensional supergravity theory developed in the 1970s, subsequently ignored, and more recently shown to be an important part of string theory.

  Entropy. A measure of the disorder of a physical system; the number of rearrangements of the ingredients of a system that leave its overall appearance intact.

  Equivalence principle. See principle of equivalence.

  Event horizon. The one-way surface of a black hole; once penetrated, the laws of gravity ensure that there is no turning back, no escaping the powerful gravitational grip of the black hole.

  Extended dimension. A space (and spacetime) dimension that is large and directly apparent; a dimension with which we are ordinarily familiar, as opposed to a curled-up dimension.

  Extremal black holes. Black holes endowed with the maximal amount of force charge possible for a given total mass.

  Families. Organization of matter particles into three groups, with each group being known as a family The particles in each successive family differ from those in the previous by being heavier, but carry the same electric and nuclear force charges.

  Fermion. A particle, or pattern of string vibration, with half a whole odd number amount of spin; typically a matter particle.

  Feynman sum-over-paths. See sum-over-paths.

  Field, Force field. From a macroscopic perspective, the means by which a force communicates its influence; described by a collection of numbers at each point in space that reflect the strength and direction of the force at that point.

  Flat. Subject to the rules of geometry codified by Euclid; a shape, like the surface of a perfectly smooth tabletop, and its higher-dimensional generalizations.

  Flop transition. Evolution of the Calabi-Yau portion of space in which its fabric rips and repairs itself, yet with mild and acceptable physical consequences in the context of string theory.

  Foam. See spacetime foam.

  Force charge. A property of a particle that determines how it responds to a particular force. For instance, the electric charge of a particle determines how it responds to the electromagnetic force.

  Frequency. The number of complete wave cycles a wave completes each second.

  Gauge symmetry. Symmetry principle underlying the quantum-mechanical description of the three nongravitational forces; the symmetry involves the invariance of a physical system under various shifts in the values of force charges, shifts that can change from place to place and from moment to moment.

  General relativity. Einstein's formulation of gravity, which shows that space and time communicate the gravitational force through their curvature.

  Gluon. Smallest bundle of the strong force field; messenger particle of the strong force.

  Grand unification. Class of theories that merge all three nongravitational forces into a single theoretical framework.

  Gravitational force. The weakest of the four fundamental forces of nature. Described by Newton's universal theory of gravity, and subsequently by Einstein's general relativity.

  Graviton. Smallest bundle of the gravitational force field; messenger particle for the gravitational force.

  Heterotic-E string theory (Heterotic E8 × E8 string theory). One of the five superstring theories; involves closed strings whose right-moving vibrations resemble those of the Type II string and whose left-moving vibrations involve those of the bosonic string. Differs in important but subtle ways from the Heterotic-O string theory.

  Heterotic-O string theory (Heterotic O(32) string theory). One of the five superstring theories; involves closed strings whose right-moving vibrations resemble those of the Type II string and whose left-moving vibrations involve those of the bosonic string. Differs in important but subtle ways from the Heterotic-E string theory.

  Higher-dimensional supergravity. Class of supergravity theories in more than four spacetime dimensions.

  Horizon problem. Cosmological puzzle associated with the fact that regions of the universe that are separated by vast distances nevertheless have nearly identical properties such as temperature. Inflationary cosmology offers a solution.

  Infinities. Typical nonsensical answer emerging from calculations that involve general relativity and quantum mechanics in a point-particle framework.

  Inflation, Inflationary cosmology. Modification to the earliest moments of the standard big bang cosmology in which universe undergoes a brief burst of enormous expansion.

  Initial conditions. Data describing the beginning state of a physical system.

  Interference pattern. Wave pattern that emerges from the overlap and the intermingling of waves emitted from different locations.

  Kaluza-Klein theory. Class of theories incorporating extra curled-up dimensions, together with quantum mechanics.

  Kelvin. A temperature scale in which temperatures are quoted relative to absolute zero.

  Klein-Gordon equation. A fundamental equation of relativistic quantum field theory.

  Laplacian determinism. Clockwork conception of the universe in which complete knowledge of the state of the universe at one moment completely determines its state at all future and past moments.

  Light clock. A hypothetical clock that measures elapsed time by counting the number of round-trip journeys completed by a single photon between two mirrors.

  Lorentz contraction. Feature emerging from special relativity, in which a moving object appears shortened along its direction of motion.

  Macroscopic. Refers to scales typically encountered in the everyday world and larger; roughly the opposite of microscopic.

  Massless black hole. In string theory, a particular kind of black hole that may have large mass initially, but that becomes ever lighter as a piece of the Calabi-Yau portion of space shrinks. When the portion of space has shrunk down to a point, the initially massive black hole has no remaining mass—it is massless. In this state, it no longer manifests such usual black hole properties as an event horizon.

  Maxwell's theory, Maxwell's electromagnetic theory. Theory uniting electricity and magnetism, based on the concept of the electromagnetic field, devised by Maxwell in the 1880s; shows that visible light is an example of an electromagnetic wave.

  Messenger particle. Smallest bundle of a force field; microscopic conveyer of a force.

  Mirror symmetry. In the context of string theory, a symmetry showing that two different Calabi-Yau shapes, known as a mirror pair, give rise to identical physics when chosen for the curled-up dimensions of string theory.

  M-theory. Theory emerging from the second superstring revolution that unites
the previous five superstring theories within a single overarching framework. M-theory appears to be a theory involving eleven spacetime dimensions, although many of its detailed properties have yet to be understood.

  Multidimensional hole. A generalization of the hole found in a doughnut to higher-dimensional versions.

  Multi-doughnut, Multi-handled doughnut. A generalization of a doughnut shape (a torus) that has more than one hole.

  Multiverse. Hypothetical enlargement of the cosmos in which our universe is but one of an enormous number of separate and distinct universes.

  Neutrino. Chargeless species of particle, subject only to the weak force.

  Neutron. Chargeless particle, typically found in the nucleus of an atom, consisting of three quarks (two down-quarks, one up-quark).

  Newton's laws of motion. Laws describing the motion of bodies based on the conception of an absolute and immutable space and time; these laws held sway until Einstein's discovery of special relativity.

  Newton's universal theory of gravity. Theory of gravity declaring that the force of attraction between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. Subsequently supplanted by Einstein's general relativity.

  Nonperturbative. Feature of a theory whose validity is not dependent on approximate, perturbative calculations; an exact feature of a theory.

  Nucleus. The core of an atom, consisting of protons and neutrons.

  Observer. Idealized person or piece of equipment, often hypothetical, that measures relevant properties of a physical system.

  One-loop process. Contribution to a calculation in perturbation theory in which one virtual pair of strings (or particles in a point-particle theory) is involved.

  Open string. A type of string with two free ends.

  Oscillatory pattern. See vibrational pattern.

  Particle accelerator. Machine for boosting particles to nearly light speed and slamming them together in order to probe the structure of matter.

  Perturbation theory. Framework for simplifying a difficult problem by finding an approximate solution that is subsequently refined as more details, initially ignored, are systematically included.