An engineer, a mathematician and a physicist walk into a universe. How many dimensions do they find?
The engineer whips out a protractor and straightedge. That’s easy, she says. With her instruments she demonstrates the trio of directions at right angles to each other: length, width and height. “Three,” she reports.
The mathematician gets out his notepad and creates a list of regular, symmetric geometric shapes with perpendicular sides. Squares have four linear edges, he notes. Cubes have six square sides. By extrapolation, hypercubes have eight cubic sides. Continuing the pattern, he realizes that he could keep going forever. “Infinity,” he says.
Finally it is the physicist’s turn. She gazes at the stars and carefully records their behavior. She determines that they attract each other through gravity, which drops off as the square of their mutual distances—an indication, she thinks, of three dimensions. However, once she derives the equation for how their light moves through space, she finds that it is best expressed in four dimensions. Then, after much thought, she tries to think of ways to describe gravity and light in a common theory, which seems to require at least ten dimensions. “Three, four, or maybe even more,” she chimes in.
Let’s see how she reached her conclusions.
In 1917, Austrian physicist Paul Ehrenfest wrote a thought-provoking piece, “In what way does it become manifest in the fundamental laws of physics that space has three dimensions?” (pdf). In the article he enumerated evidence that three dimensions are perfect for describing our world.
He noted, for example, that the stable orbits of planets in the solar system and the stationary states of electrons in atoms require inverse-squared force laws. If gravity, for instance, dropped off with the cube instead of the square of distance from the Sun, the planets would not follow steady, elliptical paths.
Let’s think of what an inverse-squared law means. Imagine a bubble that roughly encompasses a planet’s orbit. The strength of the Sun’s gravitational field at that distance is diluted over the bubble’s surface area. Surface area is proportional to radial distance squared, explaining why gravity drops off by that factor. Because a bubble, including its interior, is three-dimensional, space itself must be as well. In short, the fact that gravity tapers off with distance squared—the amount of a bubble’s surface area—implies three-dimensionality.
The universe is not just space, though. As Russian-German mathematician Hermann Minkowski demonstrated, Einstein’s special theory of relativity, postulated to explain how light moves at a constant speed relative to all observers, can best be expressed in four dimensions. Instead of considering space and time independently, he proposed a unified vision of spacetime. In his general theory of relativity, Einstein made use of the concept and described gravity using a dynamic four-dimensional model.
Light stems from electromagnetic interactions, one of the four natural forces. For many decades, physicists have been searching for methods to unite that force with the others—the strong nuclear force, weak nuclear force, and, thorniest of all, gravity—to create a single, elegant theory of fundamental forces. Two of the earliest schemes (before the strong and weak nuclear forces were identified) were independently developed by German mathematician Theodor Kaluza and Swedish physicist Oskar Klein. Though we now know that their approaches were inaccurate, each proposed to unify electromagnetism and gravity by extending general relativity by an extra dimension. Klein’s contribution best addressed the question of why such a fifth dimension would not be observable—consistent with Ehrenfest’s conclusion that space appears three-dimensional. In an idea known as compactification, Klein envisioned that the higher dimension would be rolled up into a tiny, compact loop on the order of 10-33 centimeters. Thus, while it would supply (in theory, if not in practice) a means of unification, it would be undetectable—like a curled up pill bug camouflaged as a dot on a leaf.
Klein’s contemporaries in the late 1920s, shaping the fundamentals of quantum mechanics, chose to explore the possibility of internal (pertaining to an abstract, mathematical space) dimensions, rather than physical ones that supplement spacetime. They developed their theories in Hilbert space, a mathematical construction that makes use of the infinite number of mathematical dimensions to allow for an indefinitely large assortment of quantum states. Aside from Einstein and his assistants Peter Bergmann and Valentine Bargmann, few physicists investigated the notion of unseen extra dimensions in the physical universe. (In the late 1930s and early 1940s, Einstein, Bergmann and Bargmann tried unsuccessfully to extend general relativity’s four-dimensional spacetime by an extra physical dimension to incorporate electromagnetism.)
In the 1970s and 1980s, Kaluza-Klein theory experienced a revival thanks to the emergence of superstring theory and its cousin supergravity: the idea that the fundamental components of nature are vibrating strands of energy. Mathematically, superstring theory turned out to be viable only in ten dimensions or more. Consequently, researchers began contemplating ways in which the extra six or more dimensions could be compactified.
Superstring theory evolved in the 1990s into a more general approach, called M-theory, that incorporated energetic membranes, nicknamed “branes,” along with strings. M-theory included the possibility of a large extra dimension, supplementing the ten essential dimensions in which superstrings could live. “Large” in that context meant “potentially observable,” rather than minuscule and compact.
Soon researchers realized that the large extra dimension could potentially solve a conundrum called the hierarchy problem. That dilemma involves the striking weakness of gravity compared to the other forces of nature, such as electromagnetism. A simple experiment illustrates that imbalance. Pick up a steel thumbtack with a tiny kitchen magnet, and see how its attraction overwhelms the gravitational pull of the entire earth.
In the “brane world” scenario, first proposed by physicists Nima Arkani-Hamed, Savas Dimopoulos, and Gia Dvali (a collaboration abbreviated as “ADD”), and later developed by Lisa Randall, Raman Sundrum, and others, reality consists of two branes, separated by a higher-dimensional gap called the bulk, in a configuration something like the Grand Canyon. Like timid tourists perched on a canyon rim, most particles cling to one of the branes. Consequently, the familiar physical world is situated there. Stalwart hikers that they are, gravitons, the carriers of gravity, are offered an exception and are able to explore the bulk in between. Because gravity’s agents spend much less time interacting with our familiar brane, gravity seems much weaker than the other forces.
The original ADD conjecture predicted that, when measured at fine scales , gravity should deviate subtly from a perfect inverse-squared distance relationship. However precise torsion balance experiments performed by a team led by Eric Adelberger of the University of Washington placed strict constraints on such a discrepancy down to minute levels. Nevertheless, the idea of extra dimensions has continued to flourish in various proposals for unification of the natural forces.
One of the missions of the Large Hadron Collider (LHC), the behemoth accelerator straddling the French-Swiss border, has been to test the possibility of unseen extra dimensions. Since the discovery of the Higgs Boson in 2012, completing the Standard Model of particle physics, the idea of looking at such extensions has become more central.
To establish the existence of extra dimensions with the LHC, there are three major avenues of attack. The first involves finding echo versions of existing particles, called Kaluza-Klein states. These would be like the known particles in all respects, except more massive, like overtones in music. At a proton-proton collision energy of 7 trillion electron volts, searches have been made for Kaluza-Klein gravitons, Kaluza-Klein gluons and others, so far to no avail.
Physicists are also using the LHC to search for evidence of gravitons seeping into higher dimensions. Such signals of otherwise unexplained missing energy would have to be sifted from enormous numbers of collision events, carefully ruling out a plethora of more mundane possibilities, such as escaped neutrinos.
Evidence for extra dimensions could also show up at the LHC in the form of microscopic black holes, predicted by certain higher dimensional theories. Famously, before the LHC opened, alarmists raised a fear of such objects destroying the Earth, despite calculations showing they would harmlessly decay within a tiny fraction of a second. Despite the hopes and warnings, miniature black holes have yet to be detected among the collision data of LHC experiments.
Currently, the LHC is switched off and being revamped in preparation for cranking up its collision energy almost twice as high as the previous run. In 2015 it is expected to reopen and collide protons at 13 trillion electron volts, offering the possibility of producing more massive particles and more unusual events. The upgrade will offer a greater chance to detect evidence of extra dimensions.
Engineers will marvel, no doubt, at its gleaming mechanisms, while mathematicians will be awestruck by the sheer quantity of its collected data and the powerful algorithms sifting through it. And physicists will wait eagerly for possibly the first evidence of a higher-dimensional realm beyond space and time.
Editor’s picks for further reading
Cosmos: Carl Sagan: The 4th Dimension
In this scene from the classic “Cosmos” series, Carl Sagan imagines what happens when a three-dimensional character enters a two-dimensional world.
Written in 1884, E.A. Abbott’s classic novella simultaneously satires Victorian culture and imagines life in a two-dimensional world.
NOVA: Imagining Other Dimensions
Journey from a two-dimensional “flatland” to the ten- (or more) dimensional world of superstring theory in this illustrated essay.