Now, being within an order of magnitude is certainly not compelling evidence, on its own, of equality. But a remarkable mathematical relationship does exist that made believers out of many theorists long before the appropriate experimental evidence was amassed. It turns out that general relativity implies that if the geometry of the universe is not flat, then, as the universe expands, it quickly moves farther and farther away from the mathematical equality implied by flatness. Since the universe is over ten billion years old, it is difficult to imagine how the relation between the expansion rate and the mass density could still remain so close to that for a flat universe unless the universe was, in fact, precisely flat. This puzzle was so significant that cosmologists even gave it a name: the flatness problem. In 1981 a Stanford physicist (now at MIT) named Alan Guth proposed an ingenious mechanism that would resolve this puzzle by producing, independent of its initial conditions, a flat universe today. His idea, called inflation, was that the universe underwent a rapid early period of expansion, far faster than had previously been envisaged. Like a balloon being blown up, as the universe inflated, any original curvature of space would be progressively reduced, ultimately producing a universe that was indistinguishable from a precisely flat universe. What’s more, Guth demonstrated that physical conditions that would lead to an early inflationary phase could arise naturally in so-called grand unified models of particle physics, which I shall later describe, in which the fundamental forces in nature are unified into a single force at very early times. Once Guth had shown that inflation could easily result in these models, and how it could resolve a variety of fundamental problems in cosmology beyond the flatness problem, it quickly became the basis of what is now considered the standard model of cosmology. Aside from Guth’s inflationary paradigm, there is, however, another reason a flat universe is particularly attractive, at least from a theorist’s perspective: The total gravitational energy of a flat universe is precisely zero!
How can a universe full of matter and radiation have zero total energy? While the energy associated with these quantities, in the absence of considerations of gravity, is indeed positive, it turns out that the gravitational energy of attraction between objects is negative. This is another way of saying that it takes energy to pull objects farther apart, so they have less energy if they are close together. Hence, all objects of a finite size have less gravitational energy than they would have if they were dispersed over infinitely large distances. If we define such a state in which matter is infinitely diluted as having zero energy, then all other, smaller, configurations have negative energy. If this negative energy precisely cancels the positive energy of matter and radiation in the universe, then general relativity tells us that the overall curvature of space vanishes.
Moreover, with zero net energy, the possibility that the universe itself arose spontaneously out of nothing becomes at least plausible, since one would imagine that “nothing” would also have zero energy. As Guth put it: “There is such a thing as a free lunch!” It was theoretical considerations such as these, which are primarily mathematically aesthetic, that convinced most theorists and ultimately even many observers, well in advance of the cosmic microwave background observations, that the universe was flat. In this case, as sometimes but not always happens in science, nature cooperated.
However, it was premature to slap ourselves on our collective backs and congratulate one another. For, what actually makes the universe flat is something that no one, or at least almost no one, anticipated. Perhaps the most puzzling discovery in all of physics during the past century has been the fact that the dominant form of energy in the universe is not associated with matter or radiation at all. Rather, it appears that empty space, devoid of any particles at all, carries energy—enough energy, in fact, to overwhelm, by a factor of almost three, the energy of everything else in the universe.
This energy of empty space, sometimes called “vacuum energy” or “dark energy,” is the most mysterious form of energy we know of. No one currently has a good explanation of why empty space should have precisely this amount of energy, and, as we shall see, trying to understand its nature is currently driving much of our current scientific thinking about the nature of space and time itself.
The discovery of a mysterious energy permeating all of empty space also changed everything in the way we think about cosmology. Even the original, vital connection between geometry and destiny is now gone. If empty space can possess energy, a positively curved universe need not ultimately collapse, while a negatively curved or flat universe need not expand forever. Still, as I have suggested, it could be that there might be some deeper connection between the geometry of space and its energy content, perhaps something that involves probing yet deeper into the meaning of space and time. Certainly the puzzle of dark energy is so revolutionary it motivates even extreme reconsiderations of the nature of space and time. The resolution of this mystery may not be as revolutionary as the question itself, but one never knows until one explores the possibilities. But extraordinary claims require extraordinary evidence, as Carl Sagan used to say. We shall return to this mystery later in the book. First, however, we shall explore how the collective creative imagination of the world responded to our first revolution in the physics of space and time inspired by Einstein and later Minkowski: Namely, the existence of a fourdimensional space-time continuum associated with special relativity.
C H A P T E R 7
FROM FLATLAND TO PICASSO
Ever drifting down the stream—
Lingering in the golden gleam—
Life, what is it but a dream?
—Lewis Carroll, Through the Looking Glass
While life may imitate art, it is nevertheless also true that art imitates life. One might thus wonder whether the publication of Abbott’s Flatland within a decade following Maxwell’s discovery about the nature of otherwise invisible electric and magnetic fields and less than a decade before Michelson and Morley’s experiments to probe the ether and Lorentz’s pioneering speculations about the nature of space and time was purely a coincidence. Was there something in the intellectual air at the time that suggested something revolutionary was about to occur in our understanding of nature?
In one sense the answer to this question is clearly no. It was, after all, in 1900 that Lord Kelvin uttered his famous remark that all laws of physics had already been discovered and all that remained were more and more precise measurements. Yet in spite of such hubris, scientific and mathematical puzzlement about the nature of space and time had been spilling over to the literary imagination for well over a century before Abbott wrote his story. The notion that time might somehow be considered a fourth dimension actually appeared in print as early as 1754, in an article by Jean Le Rond d’Alembert on “Dimensions” in his Encyclopédie, although he attributed the idea to a friend, possibly the French mathematician Joseph-Louise Lagrange. A hundred years later German psychologist and spiritualist Gustav Fechner wrote a satirical piece involving a “shadow man,” the shadow projection of a three-dimensional image. Interestingly, Fechner argued that such shadow figures would interpret the effects of motion perpendicular to their plane of existence (which they, of course, could not perceive as movement in space) as acting like time. Fechner’s combined interest in extra dimensions and spiritualism presaged, as we shall see, events that would unfold a half a century later.
Ultimately the notion of time as a fourth dimension was made famous within popular culture a full decade before Einstein’s special relativity and thirteen years before Minkowski clarified the dimensional relationship between space and time by none other than H. G. Wells in his classic science fiction epic, The Time Machine, published in 1895. On the very first page of this novel, Wells’s hero, the Time Traveller, has the following dialogue with an audience he has invited for the occasion:
“You must follow me carefully. I shall have to controvert one or two ideas that are almost universally accepted. The geometry, for instance, they taught you at s
chool is founded on a misconception.”
“Is not that rather a large thing to expect us to begin upon?” said Filby, an argumentative person with red hair.
“I do not mean to ask you to accept anything without reasonable ground for it. You will soon admit as much as I need from you. You know of course that a mathematical line, a line of thickness NIL, has no real existence. They taught you that? Neither has a mathematical plane. These things are mere abstractions.”
“That is all right,” said the Psychologist.
“Nor, having only length, breadth, and thickness, can a cube have a real existence.”
“There I object,” said Filby. “Of course a solid body may exist. All real things.”
“So most people think. But wait a moment. Can an INSTANTANEOUS cube exist?”
“Don’t follow you,” said Filby.
“Can a cube that does not last for any time at all, have a real existence?”
Filby became pensive. “Clearly,” the Time Traveller proceeded, “any real body must have extension in FOUR directions: it must have Length, Breadth, Thickness, and—Duration. But through a natural infirmity of the flesh, which I will explain to you in a moment, we incline to overlook this fact. There are really four dimensions, three which we call the three planes of Space, and a fourth, Time. There is, however, a tendency to draw an unreal distinction between the former three dimensions and the latter, because it happens that our consciousness moves intermittently in one direction along the latter from the beginning to the end of our lives.”
“That,” said a very young man, making spasmodic efforts to relight his cigar over the lamp, “that . . . very clear indeed.”
“Now, it is very remarkable that this is so extensively overlooked,” continued the Time Traveller, with a slight accession of cheerfulness. “Really this is what is meant by the Fourth Dimension, though some people who talk about the Fourth Dimension do not know they mean it. It is only another way of looking at Time. THERE IS NO DIFFERENCE BETWEEN TIME AND ANY OF THE THREE DIMENSIONS OF SPACE EXCEPT THAT OUR CONSCIOUSNESS MOVES ALONG IT. But some foolish people have got hold of the wrong side of that idea. You have all heard what they have to say about this Fourth Dimension?”
“I have not,” said the Provincial Mayor.
“It is simply this. That Space, as our mathematicians have it, is spoken of as having three dimensions, which one may call Length, Breadth, and Thickness, and is always definable by reference to three planes, each at right angles to the others. But some philosophical people have been asking why THREE dimensions particularly—why not another direction at right angles to the other three?—and have even tried to construct a FourDimension geometry. Professor Simon Newcomb was expounding this to the New York Mathematical Society only a month or so ago. You know how on a flat surface, which has only two dimensions, we can represent a figure of a three-dimensional solid, and similarly they think that by models of three dimensions they could represent one of four—if they could master the perspective of the thing. See?”
This passage is remarkable not merely because of Wells’s anticipation of a connection between space and time in a four-dimensional framework, but because he correctly recognized that what fascinated writers and the public alike was not a temporal fourth dimension but a spatial one. Wells also wrote several stories reminiscent of Flatland, in which he utilized four spatial dimensions as plot devices. In no fewer than four tales Wells exploited different manifestations of extra dimensions that would be borrowed by a host of future science fiction writers. These included a story involving a person being turned into his mirror image through a four-dimensional rotation, the possibility of connecting otherwise distant locations in three-dimensional space via a four-dimensional portal, the mysterious appearance and disappearance of a four-dimensional being (an angel, as it happens) traveling through our three-dimensional plane of existence, and finally an object achieving invisibility by sliding into the fourth dimension.
About 150 years earlier, around the same time as d’Alembert was writing, none other than Immanuel Kant was pondering the possibilities of extra spatial dimensions. While he may have felt that Euclidean geometry was an essential part of existence, he was much more sanguine about variations beyond our three-dimensional space, although he felt that while they could exist, they must be separate from ourselves. He discussed this possibility in his very first published work, Thoughts on the True Estimation of Living Forces, concluding: “Spaces of this kind, however, can not stand in connection with those of a quite different constitution. Accordingly such spaces would not belong to our world, but must form separate worlds.”
The German physicist and mathematician August Möbius, father of the famous one-sided Möbius strip, followed up on Kant’s earlier musings from the 1700s and came up with an interesting suggestion. He argued in 1827 that a fourth dimension would allow otherwise distinct threedimensional figures—such as a right hand and a left hand—to coincide. Namely, just as a mirror flips left and right, one could turn a right hand into a left hand by twisting it into a fourth dimension and back again. Indeed, Kant, himself, in his Prolegomena to Any Future Metaphysics (1793) wondered explicitly about how a right hand becomes a left hand when viewed in a mirror, and so two identical objects can at the same time be completely different.
The premise inherent in Flatland was that we could simply be ignorant of an ever-present fourth spatial dimension, which would appear as foreign to our intuition as a third dimension would be to a two-dimensional being. Abbott was, of course, not writing in a vacuum, and there was a swirl of activity in England in the years prior to 1884 surrounding attempts to understand physically and mathematically what a fourth dimension might be like. As I have mentioned, H. G. Wells himself wrote at least one tale in which this very issue is central. His “The Plattner Story”(1896) focuses on an individual who moves into the fourth dimension and returns with left and right inverted. Almost eighty years later, a charming rendition of this same apparent paradox was replayed in Lars Gustafsson’s tale The Death of a Beekeeper. The protagonist muses: “But, since I moved outside the normal dimensions, right and left somehow got exchanged. My right hand is now my left one, my left hand my right one.” At the same time, this transition changes his previous, pessimistic, view of our world: “Returned into the same world and see it now as a happy one. The shreds of peeled paint on the door belong to a mysterious work of art.” If only it were so. Numerous authors before Abbott had exploited two-dimensional beings as an allegory to help us imagine a fourth dimension. In England, the mathematician J. J. Sylvester wrote a popular article using them in 1869. In it he quoted from the biography of the great mathematician Gauss, in which the late mathematician was reported to have stated that he had kept several geometric questions aside, waiting to pass on so that he would have a better appreciation of four or more dimensions!
Not only was Sylvester a bold advocate of understanding four dimensions, he also firmly believed that higher dimensions actually exist, and strongly asserted an “inner assurance of the reality of transcendental space.”
Another mathematician who popularized two-dimensional beings was Charles Dodgson, known to the world as Lewis Carroll, the author of the Alice in Wonderland stories. In an 1865 story, entitled “Dynamics of a Particle,” he described a romance between a pair of linear, one-eyed animals moving along on a flat surface. When I first learned this fact I was particularly intrigued, because Through the Looking Glass (1872) was the first story I could remember that envisaged a foreign world lying right beneath our eyes. Moreover it was a world I had been fascinated with as a child—so much so that it influenced the title of this book. What if the world hiding on the other side of a mirror was real?
I have since learned, however, that Dodgson was in fact parodying the British fascination of the time with the literal idea of a fourth dimension. Dodgson’s mirror world of talking chessmen and tiger lilies may not appear to a modern reader to deal directly with such issues, but apparent
ly the psyches of nineteenth-century British readers were more attuned to his satire. At least the white queen’s memories involved both the past and the future, so time appeared to be heavily involved in the mix. Or maybe it was the queen’s propensity for believing six impossible things before breakfast that Dodgson employed to parody the fads of the time. Actually, Dodgson later became interested in the occult, and with that presumably his skeptical attitude toward extra dimensions disappeared. In the late 1870s a more sinister application of the fourth dimension appeared when a German physicist and astronomer, J. C. F. Zöllner, who in his day job (or, more appropriately, night job) actually invented a method of accurately measuring the brightness of stars, became fascinated with an American medium named Henry Slade. In séances carried out for Zöllner and others Slade performed magic tricks—such as untying a knotted cord without touching it, and transporting objects out of a sealed container—that seemed to defy explanation unless somehow he was reaching “into” an extra dimension. Like many of those who followed (including Russell Targ and Harold Puthoff, who in the 1970s claimed to have found scientific evidence for remote perception), Zöllner left scientific skepticism behind and fell for Slade’s chicanery, becoming his ardent defender and writing prolifically about Slade’s empirical demonstrations of the existence of extra dimensions. Zöllner’s fascination with Slade and the occult strikes to the heart, just as forcefully as Alice’s yearning to disappear into the mirror, of why humans have always seemed to want to believe in the possibility of extra dimensions. We seem to need somewhere beyond the world of our experience, a place that’s either better or just different. Part of this desire, I believe, arises because, while science describes the workings of the natural world, it does so without reference to “purpose,”
Hiding in the Mirror: The Quest for Alternate Realities, From Plato to String Theory (By Way of Alicein Wonderland, Einstein, and the Twilight Zone) Page 8