THE UNIVERSE UNRAVELS

Everything You Always Wanted to Know About the End of the World but Were Afraid to Ask

In the year 360 B.C. or thereabouts, Plato published Timaeus, through which Plato presents his own theory of everything, with geometry playing a central role in those ideas. And it is there, in ancient Greece, our story begins.

Aristotle argued as much in his treatise On the Heavens: "A magnitude if divisible one way is a line, if two ways a surface, and if three a body. Beyond these there is no other magnitude, because the three dimensions are all that there are." 1 In A.D. 150, Ptolemy tried to prove that four dimensions are impossible, insisting that you cannot draw four mutually perpendicular lines. His argument, however, was less a rigorous proof than a reflection of our inability both to visualize and to draw in four dimensions.

One of the first big breakthroughs in charting higher-dimensional space came courtesy of Rene Descartes, who taught us that thinking in terms of coordinates rather than pictures can be extremely productive. For with his coordinate system in hand, it became possible to use algebraic equations to describe complex, higher-dimensional geometric figures that are not readily visualized.

The great German mathematician Georg Friedrich Bernhard Riemann took off with this idea two centuries later and carried it far. In the 1850s, while working on the geometry of curved (non-Euclidean) spaces, Riemann realized that these spaces were not restricted in terms of the number of dimensions. However, Riemann's advanced mathematics had simply outpaced the physics of his era, and it takes another fifty years or so -- for the physicists, or at least one physicist in particular, to catch up.

The one who did was Albert Einstein. In developing his special theory of relativity, Einstein drew on an idea that was also being explored by the German mathematician Hermann Minkowski, namely, that time is inextricably intertwined with the three dimensions of space, forming a new geometrical construct known as spacetime. In an unexpected turn, time itself came to be seen as the fourth dimension that Riemann had incorporated decades before in his elegant equations.

Curiously, the British writer H. G. Wells had anticipated this same outcome ten years earlier in his novel The Time Machine. As explained by the Time Traveller, "there are really four dimensions, three which we call the three planes of Space, and a fourth, Time". 2 Minkowski said pretty much the same thing in a 1908 speech: "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." 3 It thus takes four coordinates to describe an event in four-dimensional spacetime (x, y, z, t).

In Einstein's theory, it takes ten numbers -- or ten fields -- to precisely describe the workings of gravity in four dimensions. The force can be represented most succinctly by taking those ten numbers and arranging them in a four-by-four, with ten of the sixteen entries are independent. Kaluza had basically taken Einstein's general theory of relativity and added an extra dimension to it by expanding the four-by-four matrix to a five-by-five one. By expanding spacetime to the fifth dimension, Kaluza was able to take the two forces known at the time, gravity and electromagnetism, and combine them into a single, unified force. To an observer in the five-dimensional world that Kaluza envisioned, those forces would be one and the same, which is what we mean by unification. But in a four-dimensional world, the two can't go together; they would appear to be wholly autonomous. You could say that's the case simply because both forces do not fit into the same four-by-four matrix. The additional dimension, however, provides enough extra elbow room for both of them to occupy the same matrix and hence be part of the same, more all-encompassing force.

In 1926 from Oskar Klein, the Swedish physicist who carried Kaluza's idea a step further. Drawing on quantum theory, Klein actually calculated the size of the compact dimension, arriving at a number that was tiny indeed -- close to the so-called Planck length, which is about as small as you can get -- around 10^30 cm in circumference. 5 In fact, he liked the idea enough to pursue Kaluza-Klein-inspired approaches off and on over the next twenty years. But ultimately, Kaluza-Klein theory was cast aside while the quantum revolution was beginning to take hold.

General relativity, the geometry-based theory that encapsulates our current understanding of gravity, has also held up extraordinarily well since Einstein introduced it in 1915, passing every experimental test it has faced. And quantum theory beautifully describes three of the known forces: the electromagnetic, weak, and strong. Predictions of the behavior of an electron in the presence of an electric field, for example, agree with measurements to ten decimal points. Unfortunately, these two very robust theories are totally incompatible. It's hardly an ideal state of affairs when your two most successful theories -- one describing large objects such as planets and galaxies, and the other describing tiny objects such as electrons and quarks -- combine to give you gibberish. Keeping them separate is not a satisfactory solution, either, because there are places, such as black holes, where the very large and very small converge, and neither theory on its own can make sense of them. "There shouldn't be laws of physics," Strominger maintains. "There should be just one law and it ought to be the nicest law around." 6

That is the promise of string theory, an intriguing tough unproven approach to unification that replaces the pointlike objects of particle physics with extended (though still quite tiny) objects called strings. Like the Kaluza-Klein approaches that preceded it, string theory assumes that extra dimensions beyond our everyday three (or four) are required to combine the forces of nature. Most versions of the theory hold that, altogether, ten or eleven dimensions (including time) are needed to achieve this grand synthesis.

String theory takes the old Kaluza-Klein idea of one hidden "extra" dimension and expands it considerably. If we were to take a detailed look at our four-dimensional spacetime, as depicted by the line in this figure, we'd see it's actually harboring six extra dimensions, curled up in an intricate though minuscule geometric space known as a Calabi-Yau manifold. More will be said about these spaces in the book The Shape of Inner Space.

I may get in trouble for saying this, but I believe that only a mathematician, would have been bold enough to think that higher-dimensional space would afford us special insight into phenomena that we've so far only managed to observe in a lower-dimensional setting. And only a computer scientist would have been bold enough to explore. Therefore comes our problem.

Reference

1. Aristotle, On the Heavens, at Ancient Greek Online Library, http://greektexts.com/library/Aristotle/On_The_Heavens/eng/print/1043.html.

2. H. G. Wells, The Time Machine (1898), available at http://www.bartleby.com/1000/1.html.

3. Abraham Pais, Subtle Is the Lord (New York: Oxford University Press, 1982), p. 152.

4. Oskar Klein, "From My Life of Physics," in The Oskar Klein Memorial Lectures, ed. Gosta Ekspong (Singapore: World Scientific, 1991), p. 110.

5. Leonard Mlodinow, Euclid’s Window (New York: Simon & Schuster, 2002), p. 231.

6. Andrew Strominger, “Black Holes and the Fundamental Laws of Nature,” lecture, Harvard University, Cambridge, MA, April 4, 2007.

-- By Dr. Shing-Tung Yau, Steven Nadis, Yingying Wu, Xiaoshi Lu, Mike Lydon, illustrated by Dr. David Xianfeng Gu, Problem statement based on Yau, Shing-Tung (2010). The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions