One could entertain the idea that the real world requires more than one, or possibly all, of the theories permitted by the mathematics.

It’s time once again to illuminate the frontiers of modern physics with movie trivia.

The film is Frequency (Dennis Quaid, Jim Caviezel, New Line Cinema, 2000). Through a ham radio mysteriously able to transmit 30 years into the past, Quaid learns from the 1999 version of his son (Caviezel) what will happen in the 1969 World Series. And the trivia question is: Who is the physicist appearing on a TV show running in the background during a scene early in the movie?

The answer is Columbia University string theorist Brian Greene, playing himself, but made up to look old and gray.

“String theory dramatically changes our understanding of space and time,” Greene tells Dick Cavett, also playing himself. “For example, it turns out that string theory requires our universe to have ten or possibly even eleven dimensions. And the strange thing is, some physicists are even pursuing the idea that there might be more than one time dimension.”

The even stranger thing is, that part of the movie wasn’t fiction. Some physicists really are pursuing the idea of a second dimension of time.

“So in addition to time as we know it,” Greene explains, “there may be a second time dimension where the universe evolves in some different manner.”

That’s about where the physics in Frequency ends, though. Greene didn’t get enough screen time to explore the ramifications of a second time dimension. But the theme of the movie does resonate with a realization that often strikes scientists and nonscientists alike: there is something mysterious about time. And all the advances of the past century in physics, while clarifying so much else about reality, have mainly deepened time’s mystery.

Just what the idea of a second time dimension means, nobody really knows. So as usual, physicists resort to concealing the mystery with a secret code. Remember, they don’t know the details of the ultimate theory of everything, so they call it M theory, for “mother of all theories.” Some theorists—Cumrun Vafa of Harvard University, for example—think M theory needs a companion. He calls the companion F theory, with the F standing for father.

F theory sounds like a way of giving fathers equal time, but in fact, time is not equal in the two theories. M theory, like all traditional theories, has just one dimension of time. F theory has two. F theory’s second time dimension is no doubt what Brian Greene was alluding to in Frequency.

Many other physicists regard the idea of a second time dimension as too bizarre, even for the movies. But perhaps it is not all that

much more bizarre than Einstein’s prediscoveries about time, nearly a century ago.

Nobody discerned more truths about the universe before they were discovered than Einstein. He foresaw the expansion of the universe and the vacuum energy or “cosmological constant.” His math predicted black holes, even though he didn’t believe in them. He realized that matter could be converted into energy and that space was curved. And he even foreshadowed the lyrics to the famous Chicago song of the 1970s: “Does anybody really know what time it is?”

Einstein not only asked the question, he answered it: No. No-body knows what time it “really” is, because there is no one real time. If you try to order events in a time sequence—first to last—your list may not match that of someone who is moving in relation to you and those events.

Not only did Einstein show that different observers would place events in a different time order, he found that the rate of time itself would change for an object in motion—the faster it moved, the more slowly time would flow. It was one of the amazing consequences about the physical world that emerged from Einstein’s 1905 special theory of relativity.

Special relativity, it seems to me, is underappreciated these days. General relativity is where the action is now, with its cutting-edge cosmological implications. All the attention to general relativity is certainly warranted. It produced many of the most profound prediscoveries of the past—black holes, and the expansion of the universe, for example—and retains many more potential prediscoveries up its sleeve, especially when you team it up with quantum mechanics.

But special relativity deserves a special place in the history of physics as well. It was, after all, the essential first step toward general

relativity. And it was a singular example of how the greatest genius of the twentieth century achieved success in prediscovering natural phenomena whose existence no one else ever suspected.

In 1905 Albert Einstein was unknown in the world of physics. Unable to secure a teaching job after earning his degree in physics from the University of Zurich, he had gone to work in the patent office in Bern as a technical expert, third class. He didn’t seem to be cut out for an academic career, anyway. His dislike for classroom discipline and his distrust of authority had caused his scholastic record to be less laudable than it might have been. While in elementary school, Einstein had shown little promise. When his father asked the school’s headmaster what profession young Albert should choose, the reply was not encouraging: “It doesn’t matter; he’ll never make a success of anything.”¹

According to Einstein’s own report, his distrust of authority originated at the age of 12 when he realized from his science readings that some Biblical stories he had been taught could not be true. At about the same time an uncle gave him an old geometry text, which young Albert devoured with considerable energy. Apparently this encounter with plane geometry was one of the few intellectually stimulating events of his youth; the only comparable experience was the gift of a pocket compass from his father when Albert was five.

Albert was no fan of the German school system, finding it suffocating and excessively rigorous. When his parents moved from Munich to Milan in 1894, he stayed behind to finish school. He soon quit, however, and followed his family to Italy just a few months before he was to have received his diploma.

After taking a few months off from education, he decided to try the Swiss system, applying in the fall of 1895 to the Federal Polytechnical Institute in Zurich. But he failed the entrance exam. So he spent a year in a Swiss high school, enabling him to get into the Zurich institute.

Einstein’s nonconforming ways continued at Zurich. He attended few lectures, preferring instead to stay in his room reading the masters of nineteenth-century physics, such as Kirchhoff, Helmholtz, Maxwell, and Hertz. He did spend a lot of time in the lab, and he passed the final exam, thanks mainly to the helpful lecture notes of his friend Marcel Grossmann.

But Einstein graduated from Zurich disgusted with the educational system. “I found the consideration of any scientific problems distasteful to me for an entire year,” he remarked. It was miraculous, he said, that “the modern methods of instruction have not yet entirely strangled the holy curiosity of inquiry.”²

In Einstein’s case at least, the system did not strangle his interest in physical science. Unfortunately, his preoccupation with physics entailed the neglect of mathematics. Einstein later commented that there existed too many branches of mathematics and no criteria by which to choose the most significant. “In physics,” he declared, “I soon learned to scent out the paths that led to the depths.”³

Upon graduation, Einstein sought employment in the laboratories of Heike Kamerlingh Onnes and Wilhelm Ostwald, but his inquiries went unanswered. For two years he struggled, making a little money by tutoring and substitute teaching. Then, in 1902, the father of a friend helped him get a job at the patent office in Bern. There, for seven years, he served as a technical expert—and there he produced some of science’s greatest insights into nature.

Isolated from the academic world of physics, Einstein’s mind was not cluttered by unnecessary knowledge or irrelevant distractions. His intuition was free to pursue its own perceptions of the physical systems he found intriguing. One of those involved a paradox he had first discerned at the age of 16, according to his own autobiographical testimony. “If I pursue a beam of light with the velocity c (velocity of light in a vacuum),” he reasoned, “I should observe such a beam of light as a spatially oscillatory electromagnetic field at rest.”⁴ Such a

phenomenon, however, seemed not to exist; nothing like it had ever been observed. It corresponded to nothing in Maxwell’s theory of electromagnetism, either. Einstein eventually concluded that there appeared to be no way to assert that a system is in a state of absolute rest. Motion, in other words, is relative.

Throughout his education at Zurich and during his first few years at the patent office, the ramifications of these realizations swirled in his mind. Finally, in 1905, a conversation with his friend Michele Besso suddenly crystallized the latent revolution. In a matter of weeks, Einstein prepared the paper that spelled out the implications of his relativity principle. He entitled it simply “On the Electrodynamics of Moving Bodies.”

Einstein built his special theory on two postulates. The first postulate: The laws governing two “reference frames,” in uniform motion with respect to each other, are the same. The second postulate: Every light ray moves through empty space with a fixed velocity c, independently of whether the ray is emitted by a body at rest or in motion. The first postulate was Einstein’s statement of the principle of relativity. Nowadays it would be regarded as a symmetry principle: the laws of nature stay the same no matter what direction or how fast you are moving. (It’s just that in the case of special relativity, motion has to be “uniform”—in a straight line with constant speed.)⁵

The second principle, Einstein declared, is contained in Maxwell’s equations. It’s part of the laws of nature that the speed of light stays the same for all observers, no matter how fast they are moving (in a straight line at constant speed). Einstein’s great insight was that these two postulates are compatible. His famous paper of 1905 began to work out the implications of that compatibility.

One significant implication concerned the notion of simultaneity. Einstein pointed out that you could find no objective point of view for deciding whether two events separated in space occurred at

precisely the same time. Whether two events were simultaneous or not depended on the motion of the observer making the judgment.

Strange new conclusions also emerged about a moving body’s mass. As a body approached the speed of light, its mass would increase; if it could attain the speed of light, its mass would become infinite. Therefore, it seemed, it would not be possible to accelerate an object with any mass at all to a velocity equal to that of light. The speed of light became a cosmic speed limit.⁶

Another curious effect of rapid motion was a foreshortening of the moving object in the direction of its motion. An observer moving along with such an object would notice nothing unusual. But an observer at rest would see a rapidly moving object appear to scrunch up—a ball, for example, would appear to flatten itself into something like a vertically oriented pancake. The amount of this scrunching increases as the velocity of light is approached. The exact degree of shrinkage can be calculated by a formula previously described by the Irish physicist George Fitzgerald and the Dutch physicist Hendrik Lorentz. (The shortening of an object in the direction of its motion is therefore referred to as the Lorentz-Fitzgerald contraction.)

It might be cheating to call this foreshortening of objects in motion a prediscovery, however. Lorentz and Fitzgerald developed their math to try to explain the famous Michelson-Morley experiment of 1887. But Einstein’s explanation for the contraction did anticipate effects not known back then—the shrinking effect has to be taken into account, for example, when analyzing the impact of fast-moving subatomic particles in accelerator experiments.

Special relativity nevertheless was rich with true prediscoveries. Most dramatic, perhaps, was Einstein’s deduction that mass and energy are equivalent, a point he spelled out in a subsequent 1905 paper. But the prediscovery most pertinent to this chapter involved a deep realization about the nature of time. A body in rapid motion, Einstein showed, experiences a slowdown in time relative to a sta-

tionary observer. Newton’s “absolute, true and mathematical time,” flowing “equably without relation to anything external” would no longer be the time of physics.

Einstein’s 1905 relativity paper showed that the same math describing the Lorentz-Fitzgerald contraction would describe the changing rate of time for objects in motion. It seems paradoxical, but Einstein mentioned it in his paper in an almost offhanded way, referring to it merely as a “peculiar consequence” of his postulates. He did not remark on how astounding it must have seemed to others who read his paper, but he illustrated the idea pretty clearly.

Consider, he wrote, two clocks, at points A and B, both at rest with respect to a coordinate system K. Make sure the clocks are synchronized and both keep good time. “If the clock at A is transported to B along the connecting line with the velocity v, then upon arrival of this clock at B the two clocks will no longer be running synchronously,” Einstein wrote. “Instead the clock that has been transported from A to B will lag . . . behind the clock that has been in B from the outset.”⁷ The precise amount of the time lag could be calculated using the Lorentz-Fitzgerald formula, with time replacing length.

Einstein went on to point out that similar reasoning applied if two clocks started out at the same spot. If one flew off and then returned, it would lag behind its stay-at-home counterpart. There was no getting around this conclusion. If Einstein’s postulates were true of nature (and they certainly seem to be), nature must play some pretty clever tricks with time. Except this trick was no illusion. It is not a case of a moving clock just turning its gears more slowly than one at rest, or of its hands encountering friction. A person traveling alongside a rapidly moving clock would notice nothing wrong with the clock. Time itself slows down for the clock, the person, and any

thing else traveling along at the same speed (with respect to a time-piece at rest). The implications became clearer when, a few years later, this clock paradox was personified into what has become known as the twin paradox.

In a 1911 lecture, Einstein spelled it out. “Whatever holds for the clock, which we introduced as a simple representation of all physical phenomena, holds also for closed physical systems of any other constitution. Were we, for example, to place a living organism in a box and make it perform the same to-and-fro motion as the clock . . . it would be possible to have this organism return to its original starting point after an arbitrarily long flight having undergone an arbitrarily small change, while identically constituted organisms that remained at rest at the point of origin have long since given way to new generations.”⁸

Therefore if one of a pair of identical twins takes off in a fast-flying rocket ship while the other twin remains homebound, the stay-at-home twin will age more rapidly, because time is slowed for the traveler.

People puzzled by this aspect of relativity still sometimes complain that the twin paradox cannot be the way it seems. If motion is relative, why can’t the stay-at-home twin pretend to be the one rapidly moving? If you restrict your analysis to special relativity, that question is hard to answer. Because the only way to test the question is for the flyboy twin to return to Earth, and to do so requires some maneuvering in space that breaks the special relativity rule about moving only in a straight line at a constant speed. In other words, the two twins do not have equivalent experiences, and the flying twin will indeed age less rapidly. (Actually, you can devise situations in which the moving twin could return younger without breaking the special relativity rules, too, but that gets a little more complicated.)⁹

In any event, the twin paradox (or as it is more properly called, the time dilation effect of special relativity) is to me a clear-cut ex-

ample of prediscovery. A century ago Newton’s absolute time seemed pretty self-explanatory to most people. The idea that the objective time of physics could slow down strikes me as utterly outside any actual physical experience or evidence. Yet Einstein’s theory, with help from the Lorentz-Fitzgerald math, revealed this aspect of the real world in advance of its discovery.

In fact, it took quite a while for that actual discovery to take place. The first really solid evidence came in the early 1940s, based on measurements of subatomic particles called muons. Muons are created in the upper atmosphere by cosmic rays striking air atoms. But muons are unstable and decay very rapidly, within a microsecond or two on average. Nevertheless many muons make it to the ground, a journey that takes much longer. The only explanation is that their rapid motion slows down their “internal clock,” giving them a long enough life to pass all the way through the atmosphere. Later experiments showed different rates of decay for muons (and also pions) rotating on the outer or inner parts of a spinning disk. Particles near the center move much more rapidly than those farther out and have a longer lifetime.

Another dramatic confirmation of time dilation came in 1972, when physicists reported a test of relativity conducted by flying atomic clocks in jet planes. The flying clocks slowed down, just as Einstein’s analysis predicted. (In that case, the time-changing effects of general relativity had to be factored into the analysis as well.)

So Einstein showed, in essence, that the Chicago song was partly right—nobody really knows what time it is. It depends on how you are moving. Neither Einstein nor Chicago, however, posed a similar question about time that a lot of physicists care about today: Does anybody really know which time it is? And they aren’t talking about time zones. They are talking about the possibility of the second time dimension that Brian Greene mentioned in Frequency.

It is now nearly a century since Einstein predicted time dilation. During that time physics has made enormous progress in understanding motion, energy, matter, and force. Much less depth of understanding has been achieved about the nature of time. I think time still holds some surprises. It’s still fair to say that nobody really knows what time is. Consequently there’s plenty of speculation about the nature of time, the arrow of time, and the possibility that time could somehow manifest itself in more than one dimension.

A few years after Einstein’s special relativity paper, his former math teacher at Zurich, Hermann Minkowski, developed the mathematical treatment of special relativity further, adding time to the three dimensions of space as an equal partner. “Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality,” Minkowski declared in 1908.¹⁰ The idea of time as a fourth dimension was not exactly new, though. H. G. Wells used it in his science fiction novel The Time Machine, and you can find earlier allusions to similar notions if you search seriously enough.¹¹ But Minkowski, using Einstein’s relativity, showed how to make the idea of time as a dimension mathematically precise. It seemed the only way to allow physics to describe events through space and time selfconsistently. So physics from then on took place in an amalgam called spacetime, three dimensions of space and one of time. Physicists describe the number of dimensions in a shorthand notation for what they call the “signature” of spacetime: (3,1) (meaning three dimensions of space, one of time).

Over the years, as we’ve seen, other physicists attempted to fool around with spacetime by adding other dimensions—but almost always just dimensions of space. In the 1980s, though, the idea of additional time dimensions began to creep into the literature, with a

mention by the famous Russian physicist Andrei Sakharov and a few others here and there.

Not that anybody noticed. I had never encountered the idea of an extra time dimension in anything remotely newsworthy. But in 1996, when writing about the then brand-new M theory, I heard about “two times” from Michael Duff at Texas A&M. In discussing the signature of spacetime preferred by M theory, Duff mentioned that it was mathematically plausible for more than one time dimension to fit into the equations. In fact, he said, he and a colleague had examined the question in 1988. Applying certain considerations of supersymmetry and other plausible restraints, they had shown which signatures of spacetime remained mathematically consistent possibilities. To their amazement, they found that some scenarios with two time dimensions seemed to make perfect sense—mathematically, at least. For years, this finding was an unremarkable curiosity, but it may turn out to have been a subtle clue to the need to recount the number of dimensions that time has to offer.

Adding a dimension of time is a new trick to teach old physicists. Most of them are happy enough to add dimensions of space. But adding a dimension of time is more controversial. Some physicists think it makes no sense. But others think it’s the only way to make sense about the latest findings on the frontiers of space and time.

In 1996, Duff could offer no good ideas for explaining what a second time dimension would mean if it did exist. But he noted that the possibility had been taken seriously by some other physicists in connection with M theory. In particular, Cumrun Vafa’s F (for father) theory described nature with 10 dimensions of space and 2 of time.

I e-mailed Vafa, inquiring about the meaning of father in this context. “My own thinking was if M theory is the ‘mother of all theories’

as one proponent of it declared, F theory would be the ‘father of all theories,’ making the relation more politically correct!” Vafa wrote back.¹² “This is not just playing with words,” he continued. “In fact the role M-theory and F-theory play in explaining new results in string theory are very much like a cooperative endeavor—(string theory is of course the offspring!).” (If you like, Vafa said, you could have the M stand for male and the F for female.)

As for the second time dimension, its meaning was not exactly clear to Vafa, or to anyone else. A second time dimension might sound like good news for people who are very busy, but the physicists I ask usually get twisted tongues when trying to explain what it would actually mean in real life.

“A hidden time dimension is much more bizarre than a hidden space dimension,” said John Schwarz. Edward Witten contended that the second time dimension in F theory merely provides a useful mathematical tool without physical significance. Andy Strominger concurred that the second time dimension seemed to be a mathematical convenience, but he wasn’t so sure what to make of it. “You put it in with the right hand and take it away with the left,” Strominger told me. “So far it’s clearly a calculational trick. But it’s a calculational trick that works so well that one suspects there’s something more behind it.”¹³

Vafa, though, objected to characterizing the second time dimension as merely mathematically useful but without physical meaning. Doubting the physical significance of F theory’s second time dimension may be ignoring lessons from history, he suggested.

“Objects which . . . resemble ‘abstract mathematical constructions’ become more ‘physical’ when we gain more insight into them,” he pointed out.¹⁴ And that’s why it makes sense to take the idea of a second time dimension seriously. The whole history of prediscovery shows that mathematical reasoning has often led the way to new physical understanding. In the mid-1960s, many scientists thought

that quarks were just convenient mathematical fictions. Now their reality is unquestioned. Even in the early days of superstring theories, the use of extra spatial dimensions seemed to some observers to be merely a way to make the math work out. Now everybody works under the assumption that the extra space dimensions are physically real.

Two time dimensions are simply what you need to make sense out of certain versions of string theory, Vafa contends. Maybe a second time direction seems odd because nobody knows where to look for it—it might come into play only in strange places, perhaps at the center of black holes. So it was too soon, he said, to dismiss the notion that a second time dimension could somehow be real. “As to what that would mean,” says Vafa, “I could only say that time will tell.”

So far time hasn’t told anybody very much. The idea of a second time dimension hasn’t grabbed the spotlight among efforts to understand M theory and the relationship of space and time to reality. But the idea hasn’t gone away, either. Papers on the second time dimension still turn up from time to time—some advocating the idea, others critiquing it. (It is, after all, one of those ideas that might turn out to be wrong.)

If it’s not wrong, though, the key to understanding a second time dimension would be in figuring out why, if it exists, nobody has noticed it. And why it doesn’t mess up the world as we know it. For as University of Pennsylvania physicist Max Tegmark has pointed out, it’s hard to reconcile a second time dimension with the existence of life.

In a paper he published in the journal Classical and Quantum Gravity, Tegmark pointed out that the existence of observers in the universe requires three qualities: complexity, stability, and predictability. That may explain why the universe has only three noticeable space dimensions. A universe with fewer than three space dimensions would

not allow enough complexity to produce life. And more than three dimensions would make stable planetary orbits impossible—a planet would either sail off into space forever or smash itself into its star. Atoms would not hold together if their particles were given the freedom of extra dimensions to roam around in. “This means that such a world cannot contain any objects that are stable over time, and thus probably cannot contain stable observers,” Tegmark wrote.¹⁵

A second dimension of time, even though logically possible, could pose equally serious problems, Tegmark believes. True, maybe a second time dimension could exist but go unnoticed. “There is no obvious reason for why an observer could not nonetheless perceive time as being one-dimensional, thereby maintaining the pattern of having ‘thoughts’ in a one dimensional succession,” Tegmark wrote. But there could still be some strange consequences. Any individual would follow a single timeline but might occasionally run into someone moving through spacetime along a different time dimension. If it were a romantic encounter, it could only lead to heartbreak. “If two . . . observers that are moving in different time-directions happen to meet at a point in spacetime, they will inevitably drift apart in separate time-directions again, unable to stay together,” Tegmark pointed out.

Such fleeting encounters would be unlikely, though, because life would be rare in a two-time universe. With an extra time direction, the subatomic particles that make up matter would easily disintegrate—in other words, everything would be radioactive. Matter would be stable only in very cold regions, which would limit life to places like Antarctica or Wisconsin. Even worse, extra time dimensions make it impossible to predict the future, Tegmark’s analysis shows. With two times, equations describing motion would differ from the usual ones in a way that would make prediction impossible.

“If an observer is to be able to make any use of its self-awareness and information-processing abilities, the laws of physics must be such

that it can make at least some predictions,” Tegmark wrote. If such predictions are impossible, then “not only would there be no reason for observers to be self-aware, but it would appear highly unlikely that information processing systems (such as computers and brains) could exist at all.”¹⁶

Since there are plenty of computers and brains around today, maybe there are no extra dimensions of time or space. On the other hand, maybe other dimensions exist but in such a way that they don’t cause trouble. In string theory, of course, the extra space dimensions are small enough, or isolated enough, to prevent serious problems. Perhaps a similar explanation applies to an extra time dimension—it could be “compactified” into closed curves, so that when you pass into another time dimension, you travel through a very brief loop, too brief to notice. Or perhaps the extra time dimension can be explained within the brane world scenario. Maybe the “bulk” space between branes contains extra space and time dimensions in which our three-brane, with one time dimension, is embedded. We don’t notice the extra time because we can’t go to the space where it operates.

Itzhak Bars, of the University of Southern California, has written a series of papers expounding on “two-time physics,” suggesting that a second time dimension might somehow be “suppressed,” sort of like the way the holograms on credit cards display what appears to be a 3-D image on a 2-D surface.

In one interesting paper, Bars and colleague Costas Kounnas of CERN argues that the familiar three-plus-one dimensional universe could have emerged from a stranger universe with up to 11 dimensions of space and as many as 3 dimensions of time. Perhaps, Bars and Kounnas propose, the big bang started only part of the universe expanding—the familiar 3 dimensions of space—while other dimensions remained small and compact. One time dimension might have taken the big-bang route while the others went in another direction.

“A plausible scenario is that one of the timelike dimensions goes along with the expanding universe and the other goes along with the compactified one,” Bars and Kounnas wrote.¹⁷

It’s hard to know what to make of ideas as wild as these. For guidance I always go back to Duff, one of those valuable sources who grasps the big picture and offers even-handed assessments. He finds the papers by Bars and most others on the issue “not very compelling.” But he expresses intrigue at an approach by the British physicist Christopher Hull. Hull’s work focuses on understanding extra times from the viewpoint of duality.

Duality, you’ll remember from Chapter 8, puts the understanding of the physical world in an entirely new perspective. It changes the very way that the notion of reality is defined. Duality is one of the most profound—and for most people, confusing—ideas of modern physics. And yet at its most basic it captures a message of utter simplicity: what’s real depends on how you look at it. Sure, a house is real, but what does it look like? That depends on whether you view it from the front or the back. Sure, an electron is real, but is it a wave or a particle? That depends on what sort of experiment you design to detect it. Sure, your theory of the universe works pretty well. But it’s not the only theory that works well, and another one, in some cases might work better, even though other times it works worse. As Niels Bohr used to say, there are two kinds of truth, trivial truths and great truths. The opposite of a trivial truth is obviously false. The opposite of a great truth is another great truth. The dualities of physics are great truths.

When it comes to time, the idea of duality may tell a great truth. Maybe the universe has two times, from one point of view. From our

point of view, though, there’s only one time. The two viewpoints are dual to each other—flip sides of a coin.

In string theory, it’s clear that duality has the power to alter the number of apparent dimensions of space, as we saw in Chapter 8. A supermembrane can wrap itself around a space dimension, kind of like Handi-Wrap around a hot dog, except there’s no hot dog, just a dimension of space. Suppose that dimension shrinks. The Handi-Wrap tightens around it, and sooner or later the Handi-Wrap looks more like a string than a membrane, and the dimension it surrounded seems to have disappeared. A theory with branes in 11 dimensions now looks like a theory of strings in 10 dimensions.

Chris Hull’s insight into the time problem is that dualities can do the same thing with time that they do with space. In switching between dual descriptions, such as when moving from the realm of weak coupling to strong coupling, maybe more than the number of space dimensions can change.

“Remarkably, it turns out that dualities can change the number of time dimensions as well,” Hull wrote in one paper, “giving rise to exotic spacetime signatures. The resulting picture is that there should be some underlying fundamental theory and that different spacetime signatures as well as different dimensions can arise in various limits.”¹⁸

So whatever the “fundamental” theory of the universe is, it should not specify one preferred spacetime signature. “Any attempt to formulate M theory or string theory as a theory in a given spacetime dimension or signature will be misleading,” Hull contends. “In particular, the theory underpinning all these theories . . . cannot at a fundamental level be a theory in 10+1 dimensions, as it has some limits which live in 9+1 dimensions and others that live in 9+2 or 6+5 dimensions.”¹⁹

In other words, the (3,1) signature of ordinary spacetime is just the one that seems most convenient from the human viewpoint. Na-

ture may encompass other signatures, some with more than one time dimension, that are just all dual versions spawned from some grander concept. There is no such thing as just “spacetime” but instead a whole class of different spacetimes, ultimately equivalent, just as the five versions of string theory turned out to be equivalent in the end.

“We have seen that different spacetimes related by dualities can define the same physics, so that the notion of spacetime geometry cannot be fundamental,” Hull asserts. Or as my friend K. C. likes to say, at the most fundamental of levels, “Space and time are toast.” Spacetime should be a derived concept, built from something else. At the moment, though, nobody knows what the something else is.

In doing away with the idea of spacetime as fundamental, Hull sees a parallel with Einstein’s relativity theories. The different frames of reference of special relativity (expanded by general relativity to include any set of spacetime coordinates) are all equally valid for describing nature. The frame of reference you use depends on the frame of reference you inhabit. In the same way, many different spacetime signatures may turn out to be equivalent, and we organize physics based on the signature that seems most sensible from our point of view.

“Two dual theories can be formulated in spacetimes of different geometry, topology and even signature and dimension,” Hull notes. “And so all these concepts must be relative rather than absolute, depending on the values of certain parameters or couplings, and such a relativity principle should be a feature of the fundamental theory that underlies all this.”²⁰ Duality may describe the second coming of Einstein’s relativity principle, in a new, more powerful form, with a vengeance.

Many of the implications of Einstein’s relativity seemed very strange, as they applied to realms of phenomena far from ordinary experience. In a similar way, the duality idea could explain the strangeness of a second time dimension. Spread all the possible

spacetimes out in a room, and you’d find that the one we inhabit is tucked off in a corner somewhere. In that corner of all possible spacetimes, an extra time dimension poses no problem. If the “real” theory has 9 space and 2 time dimensions (signature 9,2), it can be rewritten in our corner as a dual theory with 1 time and 10 space dimensions. In other corners of all possible spacetimes, where we do not live, the (9,2) signature might be much more natural.

“Some corners are stranger than others, but in any case we can only live in one corner . . . and there is no reason why other corners might not have quite unfamiliar properties,” says Hull.²¹

In other words, our grasp on reality is limited. Appearances can be deceiving. Science is about finding out what lies behind the appearances. And what lies behind might just be a second dimension of time. After all, if a second dimension of time makes the math work, there just might be something to it. As we’ve seen, math can predict some very strange things that later turn out to be discovered out there in the real world. It’s time to try to understand how math is able to do it.