From its earliest days as a science, physics has searched for unity in nature. Isaac Newton showed that the same force responsible for the fall of an apple also holds the planets in their orbits. James Clerk Maxwell combined electricity, magnetism and light into a single theory of electromagnetism; a century later physicists added the weak nuclear force to form a unified “electroweak” theory. Albert Einstein joined space and time themselves into a single spacetime continuum.

Today the biggest missing link in this quest is the unification of gravity and quantum mechanics. Einstein’s theory of gravity, his general theory of relativity, describes the birth of the universe, the orbits of planets and the fall of Newton’s apple. Quantum mechanics describes atoms and molecules, electrons and quarks, the fundamental subatomic forces, and much besides. Yet in the places where both theories should apply—where both gravity and quantum effects are strong, such as black holes—they also seem incompatible. Physicists’ best efforts to combine them into a quantum theory of gravity have failed miserably, giving answers that make no sense or no answers at all. Despite 80 years of work by generations of physicists, including a dozen or so Nobel laureates, a quantum theory of gravity remains elusive.

Ask a physicist too hard a question, and a common reply will be, “Ask me something easier.” Physics moves forward by looking at simple models that capture pieces of a complex reality. Researchers have worked on numerous such models for quantum gravity, including approximations that apply when gravity is weak or in special cases such as black holes. Perhaps the most unusual approach is to neglect a whole dimension of space and work out how gravity would operate if our universe were only two-dimensional. (Technically, physicists refer to this situation as “(2+1)-dimensional,” meaning two dimensions of space plus one of time.) The principles that govern gravity in this simplified universe might also apply to our 3-D one, thus giving us some much needed clues to unification.

The idea of dropping down a dimension has a distinguished history. Edwin Abbott’s 1884 novel Flatland: A Romance of Many Dimensions follows the adventures of “A Square,” a resident of a 2-D world of triangles, squares and other geometric figures. Although Abbott intended it as a satirical commentary on Victorian society—Flatland had a rigid class hierarchy, with linear women at the bottom and a class of circular priests at the top—Flatland also triggered a surge of interest in geometry in diverse dimensions and remains popular today among mathematicians and physicists. Researchers trying to wrap their minds around a higher-dimensional realm start by imagining what our 3-D world would look like to A Square [see “Mathematical Games,” by Martin Gardner; Scientific American, July 1980]. Flatland has also inspired physicists studying materials such as graphene that really do behave like 2-D spaces [see “Carbon Wonderland,” by Andre K. Geim and Philip Kim; Scientific American, April 2008].

The first studies of Flatland gravity, made in the early 1960s, were a letdown. A 2-D space literally would not have enough room for changes in the gravitational field to propagate. In the late 1980s, however, the subject had a renaissance as researchers realized that gravity works in unexpected ways. It would still sculpt the overall shape of space and even create black holes. Flatland gravity has been a case study in lateral thinking, letting us subject some of our speculative ideas, such as the so-called holographic principle and the emergence of time from timelessness, to a rigorous mathematical test.

Time Management
When physicists seek to develop a quantum theory of a force, we take the corresponding classical theory as our starting point and build on it. For gravity, that means general relativity. And there the trouble starts. General relativity involves a complex system of 10 equations, each with up to thousands of terms. We cannot solve these equations in their full generality, so we face a daunting task in formulating their quantum version. But the mystery of why quantum gravity is so elusive is deeper still.

According to general relativity, the thing we call “gravity” is actually a manifestation of the shape of space and time. Earth orbits the sun not because some force tugs on it but because it is moving along the straightest possible path in a spacetime that has been warped by the sun’s mass. Uniting quantum mechanics and gravity means somehow quantizing the structure of space and time itself.

That may not sound so challenging. Yet a cornerstone of quantum mechanics is the Heisenberg uncertainty principle, the idea that physical quantities are inherently fuzzy—fluctuating randomly and having no definite values unless they are observed or undergo an equivalent process. In a quantum theory of gravity, space and time themselves fluctuate, shaking the scaffolding on which the rest of physics is built. Without a fixed spacetime as the background, we do not know how to describe positions, rates of change or any of the other basic quantities of physics. Simply put, we do not know what a quantum spacetime means.

These general obstacles to conceptualizing quantized spacetime show up in several specific ways. One is the notorious “problem of time.” Time is fundamental to our observed reality. Almost every theory of physics is ultimately a description of the way some piece of the universe changes in time. So we physicists had better know what “time” means, and the embarrassing truth is that we do not.

To Newton, time was absolute—standing outside nature, affecting matter but unaffected by it. The usual formulations of quantum mechanics accept this idea of an absolute time. Relativity, however, dethroned absolute time. Different observers in relative motion disagree about the passage of time and even about whether two events are simultaneous. A clock—as well as anything else that varies in time—runs more slowly in a strong gravitational field. No longer merely an external parameter, time is now an active participant in the universe. But if there is no ideal clock sitting outside the universe and determining the pace of change, the passage of time must arise from the internal structure of the universe [see “Is Time an Illusion?” by Craig Callender; Scientific American, June 2010]. But how? It is hard to even know where to start.

The problem of time has a less famous cousin, the problem of observables. Physics is an empirical science; a theory must make verifiable predictions for observable quantities. In ordinary phys­ics, these quantities are ascribed to specific locations: the strength of the electric field “here” or the probability of finding an electron “there.” We label “here” or “there” with the coordinates x, y and z, and our theories predict how observables depend on the values of these coordinates.

Yet according to Einstein, spatial coordinates are arbitrary, human-made labels, and in the end the universe does not care about them. If you cannot identify a point in spacetime objectively, then you cannot claim to know what is going on at it. Charles Torre of Utah State University has shown that a quantum theory of gravity can have no purely local observables—that is, observables whose values depend on only a single point in spacetime. So scientists are left with nonlocal observables, quantities whose values depend on many points at once. In general, we do not even know how to define such objects, much less use them to describe the world we observe.

A third problem is how the universe came into being. Did it pop into existence from nothing? Did it split off a parent universe? Or did it do something else entirely? Each possibility poses some difficulty for a quantum theory of gravity. A related problem is a perennial favorite of science-fiction writers: wormholes, which form shortcuts between locations in space or even in time. Physicists have thought seriously about this idea—in the past 20 years they have written more than 1,000 journal articles on wormholes—without settling the question whether such structures are possible.

A final set of questions revolves around the most mysterious beasts known to science: black holes. They may offer our best window into the ultimate nature of space and time. In the early 1970s Stephen Hawking showed that black holes should glow like a hot coal—emitting radiation with a so-called blackbody spectrum. In every other physical system, temperature reflects the underlying behavior of microscopic constituents. When we say a room is hot, what we really mean is that the molecules of air inside it are moving energetically. For a black hole the “molecules” must be quantum-gravitational. They are not literally molecules but some unknown microscopic substructure—what a physicist would call “degrees of freedom”—that must be capable of changing. No one knows what they truly are.

An Unattractive Model
At first glance, Flatland seems an unpromising place to seek answers to these questions. Abbott’s Flatland had many laws, but a law of gravity was not among them. In 1963 Polish physicist Andrzej Staruszkiewicz worked out what that law might be by applying general relativity. He found that a massive object in Flatland would bend the 2-D space around it into a cone, like a party hat made by twisting a flat piece of paper. A small object passing the apex of this cone would find its path deflected, much as the sun bends a comet’s path in our universe. In 1984 Stanley Deser of Brandeis University, Roman Jackiw of the Massachusetts Institute of Technology and Gerard ’t Hooft of Utrecht University in the Netherlands worked out how quantum particles would move through such a space.

This geometry would be much simpler than the complicated pattern of curvature that gravity causes in our 3-D universe. Flatland would lack the equivalent of Newton’s law of attraction; instead the strength of the force would depend on objects’ velocities, and two bodies at rest would not be pulled toward each other. This simplicity is appealing. It suggests that quantizing Staruszkiewicz’s theory would be easier than quantizing full-blown general relativity in 3-D. Unfortunately, the theory is too simple: nothing is left to quantize. A 2-D space has no room for an important element of Einstein’s theory: gravitational waves.

Consider the simpler case of electromagnetism. Electric and magnetic fields are produced by electric charges and currents. As Maxwell showed, these fields can detach themselves from their sources and move freely as light waves. In the quantum version of Maxwell’s theory, the waves become photons, the quanta of light. In the same way, the gravitational fields of general relativity can detach themselves from their sources and become freely propagating gravitational waves, and physicists widely assume that a quantum theory of gravity will contain particles called gravitons that do the traveling.

A light wave has a polarization: its electric field oscillates in a direction perpendicular to its direction of motion. A gravitational wave also has a polarization, but the pattern is more complicated: the field oscillates not in one but in two directions perpendicular to its direction of motion. Flatland has no room for this behavior. Once the direction of motion is fixed, only one perpendicular direction remains. Gravitational waves and their quantum counterparts, gravitons, simply cannot be squeezed into two dimensions of space.

Despite occasional sparks of interest, Staruszkiewicz’s discovery languished. Then, in 1989, Edward Witten of the Institute for Advanced Study in Princeton, N.J., stepped in. Witten, widely considered the world’s leading mathematical physicist, had been working on a special class of fields in which waves do not propagate freely. When he realized that 2-D gravity fit into this class, he added the crucial missing ingredient: topology.

What Witten pointed out was that even if gravity cannot propagate as waves, it can still have a dramatic effect on the overall shape of space. This effect does not arise when Flatland is just a plane; it requires a more complicated topology. When an ice sculpture melts away, the details become muted, but certain features such as holes tend to last. Topology describes these features. Two surfaces have the same topology if one can be smoothly deformed into the other without cutting, tearing or gluing. For instance, a hemisphere and a disk share the same topology: stretching the hemisphere by pulling on its perimeter yields a disk. A sphere has another topology: to turn it into a hemisphere or disk, you would need to snip out a piece. A torus, like the surface of a doughnut, has yet another. The surface of a coffee cup has the same topology as a torus: the handle looks like a torus, and the rest of the cup can be smoothed out without cutting or tearing—thus, the old mathematician’s joke that a topologist cannot tell a doughnut from a coffee cup.

Although tori look curved, when you consider their internal geometry rather than their shape as seen from the outside, they can actually be flat. What makes a torus a torus is the fact you can make a full loop around it in two separate directions: through the hole or around the rim. This feature will be familiar to anyone who has played a 1980s-era video game where a combatant exiting the right side of the screen reenters on the left. The screen is flat: it obeys the rule of plane geometry, such as the fact that parallel lines never meet. Yet the topology is toroidal.

In fact, an infinite family of such tori exist—all flat but all distinct, labeled by a parameter called the modulus. What gravity in a toroidal universe does is to cause the modulus to evolve in time. A torus starts as a line at the big bang and opens up to assume an ever more square-shaped geometry as the universe expands. Starting with Witten’s results, I showed that this process could be quantized, turning the classical theory of gravity into a quantum one. Quantum gravity in Flatland is a theory not of gravitons but of shape-shifting tori. That view marks a shift in the usual picture of quantum theory as a theory of the very small. Quantum gravity in two dimensions is, in fact, a theory of the entire universe as a single object. This insight gives us a rich enough model to explore some of the fundamental conceptual problems of quantum gravity.

Finding the Time
Flatland gravity demonstrates, for example, how time might emerge from a fundamentally timeless reality. In one formulation of the theory, the entire universe is described by a single quantum wave function, similar to the mathematical device that physicists routinely use to describe particles and atoms. This wave function does not depend on time, because it already includes all time, past, present and future, in one package. Somehow this “timeless” wave function gives rise to the change we observe in the world. The trick is to remember Einstein’s aphorism that time is what is measured by a clock. Time does not stand outside the universe; it is determined by a subsystem that is correlated with the rest of the universe, just as a wall clock is correlated with Earth’s rotation.

The theory offers many different clock options, and our choice defines what we mean by “time.” A Square can define time by using the readings of atomic clocks in satellites, like those in the GPS. He can label time by the lengths of curves extending from the big bang, by the size of his expanding universe, or by the amount of redshift caused by its expansion. Once he has made such a choice, all other physical observables change with clock time. The modulus of the torus universe is correlated with its size, for instance, and A Square perceives this as a universe evolving in time. The theory thus bootstraps time from a timeless universe. These ideas are not new, but quantum gravity in Doughnutland has at last given us a setting in which we can do the math and check that the picture does not just look pretty but really works. Some of the definitions of time have intriguing consequences, such as implying that space can be creased.

As for the problem of observables, Doughnutland gives us a set of objectively measurable quantities—namely, the moduli. The twist is that these quantities are nonlocal: they do not reside at specific locations but describe the structure of the whole space. Anything that A Square measures is ultimately a proxy for these nonlocal quantities. In 2008 Catherine Meusburger, now at the University of Erlangen-Nürnberg in Germany, showed how these moduli relate to real cosmological measurements such as time delays and redshifts for beams of light. I have shown how they relate to objects’ motion.

Flatland gravity offers good news for fans of wormholes: at least one formulation of the theory permits the topology of space to change. A Square could go to bed tonight in Sphereland and wake up tomorrow in Doughnutland, which is equivalent to creating a shortcut between two distant corners of the universe. In some versions of the theory, we can describe the creation of the universe out of nothing, the ultimate change in topology.

On the Edge of Space
Because gravity in flatland is stunted, it used to be common knowledge among experts in the field (me included) that 2-D black holes were impossible. In 1992, though, three physicists—Máximo Bañados, now at the Pontifical Catholic University of Chile in Santiago, and Claudio Bunster (then Claudio Teitel­boim) and Jorge Zanelli, both at the Center for Scientific Studies in Valdivia, Chile—shocked the world, or at least our little corner of it, by showing that the theory does allow black holes, as long as the universe has a certain type of dark energy.

A so-called BTZ black hole is very much like a real black hole in our own universe. Formed from matter collapsing under its own weight, it is surrounded by an event horizon, a one-way barrier from which nothing can escape. To an observer who remains on the outside, the event horizon looks like an edge of the universe: any object that falls through the horizon is completely cut off from us. Per Hawking’s calculations, A Square should see it glow at a temperature that depends on its mass and spin.

That result presents a puzzle. Lacking gravitational waves or gravitons, Flatland gravity should also lack the gravitational degrees of freedom that would explain black hole temperature. Yet they sneak in anyway. The reason is that the event horizon itself provides some additional structure that empty 2-D space lacks. The horizon exists at a certain location, which, mathematically, augments the raw theory with some additional quantities. Vibrations that wiggle the horizon provide degrees of freedom. Remarkably, we find that they exactly reproduce Hawking’s results.

Because the degrees of freedom are features of the horizon, they reside, in a sense, on the edge of Flatland itself. So they are a concrete realization of a fascinating proposal about the nature of quantum gravity, the holographic principle. This principle suggests that dimension may be a fungible concept. Just as a hologram captures a three-dimensional image on a flat 2-D film, many physicists speculate that the physics of a d-dimensional world can be completely captured by a simpler theory in d–1 dimensions. In string theory—a leading effort to unify general relativity and quantum mechanics—this idea led in the late 1990s to a novel approach for creating a quantum theory of gravity [see “The Illusion of Gravity,” by Juan Maldacena; Scientific American, November 2005].

Flatland gravity provides a simplified scenario to test that approach. Just over four years ago Witten and Alexander Maloney, now at McGill University, again surprised the physics world by suggesting that the holographic predictions appear to fail for the simplest form of 2-D gravity. The theory, they found, seemed to predict impossible thermal properties for black holes. This unexpected result suggests that gravity is an even more subtle phenomenon than we had suspected, and the response has been a fresh surge of Flatland research. Perhaps gravity simply does not make sense by itself but must work in partnership with other kinds of forces and particles. Perhaps Einstein’s theory needs to be revised. Perhaps we need to find a way to put back some local degrees of freedom. Perhaps the holographic principle does not always hold. Perhaps space, like time, is not a fundamental ingredient of the universe. Whatever the answer, Flatland gravity has pointed us in a direction we might not otherwise have taken.

Although we cannot make a real 2-D black hole, we might be able to test some of the predictions of the Flatland model experimentally. Several laboratories around the world are working on 2-D analogues of black holes. For example, a fluid flowing faster than the speed of sound produces a sonic event horizon from which sound waves cannot escape. Experimenters have also built 2-D black holes using electromagnetic waves confined to surfaces [see box above]. Such analogues should also exhibit a quantum glow in much the same way a black hole does [see “Hawking Was Right (Probably),” by John Matson; Scientific American, December 2010].

Quantum gravity in Flatland began as a playground for physicists, a simple setting in which to explore ideas about real-world quantum gravity. It has already taught us valuable lessons about time, observables and topology that are carrying over to real 3-D gravity. The model has surprised us with its richness: the unexpectedly important role of topology, its remarkable black holes, its strange holographic properties. Perhaps soon we will fully understand what it is like to be a square living in a flat world.