Physics for the 21st Century
String Theory and Extra Dimensions
This unit continues our movement from an experimentally proven understanding of nature’s four fundamental forces to the theoretical effort to develop a “theory of everything” that brings all four forces under the same conceptual umbrella. The most prominent aspect of that effort is the family of string theories that envision the basic units of matter as minuscule stretches of threadlike strings rather than point particles. Introduced in the mid-1970s, the string concept has stimulated a great deal of theoretical excitement even though it has no connection to experiment so far. The unit introduces string theory in the context of quantum gravity and outlines its inherent multidimensional nature; the most promising approach involves a total of ten dimensions. The unit then covers the relationship of string theory to particle physics and introduces the idea of “branes,” related to strings. Next, the unit focuses on cosmological issues arising from our understanding of the Big Bang, outlines the way in which the concept of rapid inflation very early in the universe can solve some major issues, and details links between string theory and cosmic inflation. Finally, the unit summarizes the understanding that string theory brings to fundamental understanding of gravity.
Anti-de Sitter/Conformal Field Theory (AdS/CFT) is a mathematical relationship between two separate descriptions of the same physics. According to AdS/CFT, a string theory in a region of Anti-de Sitter (AdS) space is equivalent to a conformal field theory (CFT) on the boundary of that region. Anti-de Sitter space has negative curvature (a two-dimensional plane is curved in a saddle shape rather than flat), and is one of the simplest geometries in which the equations of general relativity can be solved. A conformal field theory is the type of field theory used in the Standard Model. Although AdS/CFT describes an artificially simple situation—we appear to live in flat space, not Anti-de Sitter space—the mathematical correspondence between the two descriptions of physics has allowed relatively straightforward field theory calculations to shed light on problems associated with the quantum mechanics of black holes. AdS/CFT has also been used the other way, with black hole calculations providing insight into complicated particle collisions and condensed matter systems that are difficult to understand with the conventional field theory approach.
A black hole is a region of space where gravity is so strong that nothing can escape its pull. Black holes have been detected through their gravitational influence on nearby stars and through observations of hot gas from surrounding regions accelerating toward them. These black holes are thought to have formed when massive stars reached the end of their cycle of evolution and collapsed under the influence of gravity. If a small volume of space contains enough mass, general relativity predicts that spacetime will become so highly curved that a black hole will form.
A blackbody is an object that absorbs all incident electromagnetic radiation and re-radiates it after reaching thermal equilibrium. The spectrum of light emitted by a blackbody is smooth and continuous, and depends on the blackbody’s temperature. The peak of the spectrum is higher and at a shorter wavelength as the temperature increases.
In string theory, branes are fundamental objects that exist in a specific number of spatial dimensions. The “p” in p-brane stands for the number of dimensions that brane has. For example, a string is a 0-brane, a membrane is a 2-brane, and we could live on a 3-brane.
In string theory, a closed string forms a loop. Unlike open strings, closed strings are not attached to other objects; however, a closed string can be broken apart to form an open string. Open strings and closed strings have different properties, and give rise to different sets of fundamental particles.
In string theory, the term compactification refers to how an extra dimension is made small enough that we cannot perceive it. The three spatial dimensions we are familiar with from daily life are essentially infinite, while compactified dimensions are curled up, and have a finite size that ranges from a few microns (10-6 m) down to the Planck length.
cosmic microwave background
The cosmic microwave background (CMB) radiation is electromagnetic radiation left over from when atoms first formed in the early universe, according to our standard model of cosmology. Prior to that time, photons and the fundamental building blocks of matter formed a hot, dense soup, constantly interacting with one another. As the universe expanded and cooled, protons and neutrons formed atomic nuclei, which then combined with electrons to form neutral atoms. At this point, the photons effectively stopped interacting with them. These photons, which have stretched as the universe expanded, form the CMB. First observed by Penzias and Wilson in 1965, the CMB remains the focus of increasingly precise observations intended to provide insight into the composition and evolution of the universe.
A cosmic string is a one-dimensional topological defect stretching across the universe, essentially an extremely thin, extremely dense line in space that would deform spacetime around it according to general relativity. Cosmic strings have been predicted by various theories, but never detected. It is possible that if the period of inflation in the early universe ended in a collision between a brane and an anti-brane, cosmic strings were produced in the process.
The cosmological constant is a constant term that Einstein originally included in his formulation of general relativity. It has the physical effect of pushing the universe apart. Einstein’s intent was to make his equations describe a static universe. After astronomical evidence clearly indicated that the size of the universe is changing, Einstein abandoned the cosmological constant though other astrophysicists, such as Georges Lemaître and Sir Arthur Stanley Eddington, thought it might be the source of cosmic expansion. The cosmological constant is a simple explanation of dark energy consistent with the observations; however, it is not the only possible explanation, and the value of the cosmological constant consistent with observation is over 60 orders of magnitude different from what theory predicts.
A black hole’s event horizon is the point of no return for matter falling toward the black hole. Once matter enters the event horizon, it is gravitationally bound to the black hole and cannot escape. However, an external observer will not see the matter enter the black hole. Instead, the gravitational redshift due to the black hole’s strong gravitational field causes the object to appear to approach the horizon increasingly slowly without ever going beyond it. Within the event horizon, the black hole’s gravitational field warps spacetime so much that even light cannot escape.
General relativity is the theory Einstein developed to reconcile gravity with special relativity. While special relativity accurately describes the laws of physics in inertial reference frames, it does not describe what happens in an accelerated reference frame or gravitational field. Since acceleration and gravity are important parts of our physical world, Einstein recognized that special relativity was an incomplete description and spent the years between 1905 and 1915 developing general relativity. In general relativity, we inhabit a four-dimensional spacetime with a curvature determined by the distribution of matter and energy in space. General relativity makes unique, testable predictions that have been upheld by experimental measurements, including the precession of Mercury’s orbit, gravitational lensing, and gravitational time dilation. Other predictions of general relativity, including gravitational waves, have not yet been verified. While there is no direct experimental evidence that conflicts with general relativity, the accepted view is that general relativity is an approximation to a more fundamental theory of gravity that will unify it with the Standard Model. See: gravitational lensing. gravitational time dilation, gravitational wave, precession, spacetime, special relativity, Standard Model.
Gravitational lensing occurs when light travels past a very massive object. According to Einstein’s theory of general relativity, mass shapes spacetime and space is curved by massive objects. Light traveling past a massive object follows a “straight” path in the curved space, and is deflected as if it had passed through a lens. Strong gravitational lensing can cause stars to appear as rings as their light travels in a curved path past a massive object along the line of sight. We observe microlensing when an object such as a MACHO moves between the Earth and a star. The gravitational lens associated with the MACHO focuses the star’ light, so we observe the star grow brighter then dimmer as the MACHO moves across our line of sight to the star.
The graviton is the postulated force carrier of the gravitational force in quantum theories of gravity that are analogous to the Standard Model. Gravitons have never been detected, nor is there a viable theory of quantum gravity, so gravitons are not on the same experimental or theoretical footing as the other force carrier particles.
The ground state of a physical system is the lowest energy state it can occupy. For example, a hydrogen atom is in its ground state when its electron occupies the lowest available energy level.
The hierarchy problem in theoretical physics is the fact that there appear to be two distinctly different energy scales in the universe for reasons that are not understood. The first energy scale, called the “electroweak scale,” is associated with everything except gravity. The electroweak scale is set by the mass of the W and Z bosons at around 100 GeV, and determines the strength of the strong, electromagnetic, and weak interactions. The second is the Planck scale, at 1019 GeV, which is associated with gravitational interactions. Another way of stating the hierarchy problem is to ask why gravity is 39 orders of magnitude weaker than the other fundamental forces of nature.
Inflation is a period of exponential expansion thought to have occurred around 10-36 seconds after the universe began. During this period, which lasted for a few million Planck times, the universe expanded by a factor of at least 1025, smoothing out temperature and density fluctuations to produce the nearly uniform universe we observe today. Although the mechanism driving inflation is still not understood, evidence from the cosmic microwave background supports its existence.
The inflaton is a hypothetical scalar field that could drive the period of inflation that took place in the early universe.
The term “nucleosynthesis” refers either to the process of forming atomic nuclei from pre-existing protons and neutrons or to the process of adding nucleons to an existing atomic nucleus to form a heavier element. Nucleosynthesis occurs naturally inside stars and when stars explode as supernovae. In our standard model of cosmology, the first atomic nuclei formed minutes after the Big Bang, in the process termed “Big Bang nucleosynthesis.”
In string theory, an open string has two distinct ends. Open strings can have one end attached to another object like a brane, and the two ends of an open string can connect to form a closed string. Open strings and closed strings have different properties, and give rise to different sets of fundamental particles.
The Planck length is the fundamental unit of length used in high energy physics, and is a combination of Planck’s constant, Newton’s constant of universal gravitation, and the speed of light. The Planck length is approximately 1.6 x 10-35 m.
The Planck mass is the fundamental unit of mass used in high energy physics, and is a combination of Planck’s constant, Newton’s constant of universal gravitation, and the speed of light. The Planck mass is approximately 2.2 x 10-8 kg.
The Planck time is the time it takes light to travel one Planck length, and is considered the fundamental unit of time in high energy physics. The Planck time is approximately 5.4 x 10-44 seconds.
Potential energy is energy stored within a physical system. A mass held above the surface of the Earth has gravitational potential energy, two atoms bound in a molecule have chemical potential energy, and two electric charges separated by some distance have electric potential energy. Potential energy can be converted into other forms of energy. If you release the mass, its gravitational potential energy will be converted into kinetic energy as the mass accelerates downward. In the process, the gravitational force will do work on the mass. The force is proportional to the rate at which the potential energy changes. It is common practice to write physical theories in terms of potential energy, and derive forces and interactions from the potential.
Quantum chromodynamics, or QCD, is the theory that describes the strong nuclear force. It is a quantum field theory in which quarks interact with one another by exchanging force-carrying particles called “gluons.” It has two striking features that distinguish it from the weak and electromagnetic forces. First, the force between two quarks remains constant as the quarks are pulled apart. This explains why single quarks have never been found in nature. Second, quarks and gluons interact very weakly at high energies. QCD is an essential part of the Standard Model and is well tested experimentally; however, calculations in QCD can be very difficult and are often performed using approximations and computer simulations rather than solved directly.
In the context of cosmology, the term recombination refers to electrons combining with atomic nuclei to form atoms. In our standard model of cosmology, this took place around 390,000 years after the Big Bang. Prior to the time of recombination, the universe was filled with a plasma of electrically charged particles. Afterward, it was full of neutral atoms.
A scalar field is a smoothly varying mathematical function that assigns a value to every point in space. An example of a scalar field in classical physics is the gravitational field that describes the gravitational potential of a massive object. In meteorology, the temperature and pressure distributions are scalar fields. In quantum field theory, scalar fields are associated with spin-zero particles. All of the force-carrying particles as well as the Higgs boson are generated by scalar fields.
Singularity is a mathematical term that refers to a point at which a mathematical object is undefined, either because it is infinite or degenerate. A simple example is the function 1/x. This function has a singularity at x = 0 because the fraction 1/0 is undefined. Another example is the center of a black hole, which has infinite density. In our standard model of cosmology, the universe we live in began as a spacetime singularity with infinite temperature and density.
Einstein developed his theory of special relativity in 1905, 10 years before general relativity. Special relativity is predicated on two postulates. First, the speed of light is assumed to be constant in all inertial frames. Second, the laws of physics are assumed to be the same in all inertial frames. An inertial frame, in this context, is defined as a reference frame that is not accelerating or in a gravitational field. Starting from these two postulates, Einstein derived a number of counterintuitive consequences that were later verified by experiment. Among them are time dilation (a moving clock will run slower than a stationary clock), length contraction (a moving ruler will be shorter than a stationary ruler), the equivalence of mass and energy, and that nothing can move faster than the speed of light. See: general relativity, spacetime.
The strong interaction, or strong nuclear force, is one of the four fundamental forces of nature. It acts on quarks, binding them together into mesons. Unlike the other forces, the strong force between two particles remains constant as the distance between them grows, but actually gets weaker when the particles get close enough together. This unique feature ensures that single quarks are not found in nature. True to its name, the strong force is a few orders of magnitude stronger than the electromagnetic and weak interactions, and many orders of magnitude stronger than gravity.
Topology is the mathematical study of what happens to objects when they are stretched, twisted, or deformed. Objects that have the same topology can be morphed into one another smoothly, without any tearing. For example, a donut and a coffee cup have the same topology, while a beach ball is in a different topological category.
In string theory, a winding mode is a distinct way in which a string can wrap around a compactified extra dimension. If we imagine a single extra dimension compactified into a circle, the simplest winding mode is for the string to wind once around the circle.
Content Developer: Shamit Kachru
Shamit Kachru is a professor of physics at Stanford University. His research interests include string theory and quantum field theory, and their applications in cosmology, condensed matter physics, and elementary particle theory. He has made central contributions to the study of compactifications of string theory from ten to four dimensions, especially in the exploration of mechanisms that could yield string models of dark energy or cosmic inflation. He has also made notable contributions to the discovery and exploration of string dualities, to the study of models of supersymmetry breaking in string theory, and to the construction of calculable dual descriptions of strongly coupled particle physics and condensed matter systems using the AdS/CFT correspondence.
Kachru earned an undergraduate degree from Harvard University in 1990 and a doctorate in physics from Princeton University in 1994. He has been a junior fellow in the Harvard Society of Fellows, and is a recipient of a Department of Energy Outstanding Junior Investigator Award, an Alfred P. Sloan Foundation Fellowship, a David and Lucile Packard Foundation Fellowship, the Bergmann Memorial Award, and the ACIPA Outstanding Young Physicist Prize. He has served on the faculties of University of California, Berkeley and the Kavli Institute for Theoretical Physics at the University of California, Santa Barbara, and has been a professor at Stanford since 1999.
Featured Scientist: Juan Maldacena
Juan Maldacena is a theoretical physicist and professor at the Institute for Advanced Study in Princeton, NJ. His work focuses on string theory, quantum gravity, and quantum field theory that concentrates on black holes. He is best known for developing a concrete realization of the holographic principle that argues that gravitational physics in certain spacetimes is equivalent to a quantum field theory on the boundary of these spacetimes. He is currently working on aspects of the gauge/gravity duality, or holography.
Maldacena was born in Buenos Aires, Argentina, and completed his undergraduate education there. He received his Ph.D. from Princeton University in 1996. In 1997 he was a visiting professor at Harvard University, and, in 1998 he served at Harvard as the Thomas D. Cabot Associate Professor. He became professor of physics at Harvard in 1999. Since 2001 he has served as a faculty member at the Institute for Advanced Study.
Featured Scientist: S.-H. Henry Tye
S.-H. Henry Tye is the Horace White Professor of Physics at Cornell University. He is a particle theorist working in quantum field theory, string theory, and cosmology. His recent research interest includes the interplay between string theory and the inflationary universe.
4.2 String Theory and Extra Dimensions — Video
In the 20th century, twin breakthroughs, quantum mechanics and general relativity, provided fresh insight into phenomena at the sub-atomic and cosmological scales, respectively. Yet physicists are still struggling to develop a consistent theory that bridges quantum mechanics and gravity. One approach to "quantum gravity" is string theory: a mathematical description of particles and forces at scales 1031 times smaller than a proton. String theory attempts to link particle physics, the forces of nature, and the earliest moments of the universe into a fresh theoretical framework. Hard evidence remains elusive, but theorists are applying string theory to important problems, such as explaining the exponential growth of the early universe and the paradoxes introduced to physics by the existence of black holes.
Supplementary: Unit 4: String Theory and Extra Dimensions — Printable Online Text
Supplemental resource for educators and students