Warped Passages by Lisa Randall Read Free Book Online Page B

quickly loop back on themselves, like a tightly wound spool of thread. No two objects could be separated very far along a compactified dimension; any attempt at a long-distance excursion would instead turn into a journey that went round and round, like the Dodo’s laps. Such compactified dimensions could be so small that we wouldn’t ever notice their existence. Indeed, we’ll see that if tiny rolled-up dimensions exist, they will be quite a challenge to detect.
    Rolled-up Dimensions in Physics
    String theory, the most promising candidate for a theory combining quantum mechanics and gravity, gives a concrete reason to think about extra dimensions: the only coherent versions of string theory that we know of are laden with these surprising appendages. However, although the arrival of string theory in the physics world improvedthe respectability of extra dimensions, the idea of extra dimensions originated much earlier.
    Back in the early twentieth century, Einstein’s theory of relativity opened the door to the possibility of extra dimensions of space. His theory of relativity describes gravity, but it doesn’t tell us why we experience the particular gravity we do. Einstein’s theory does not favor any particular number of spatial dimensions. It works equally well for three or four or ten. Why, then, do there seem to be only three?
    In 1919, close on the heels of Einstein’s theory of general relativity (completed in 1915), the Polish mathematician Theodor Kaluza recognized this possibility in Einstein’s theory and boldly proposed a fourth spatial dimension, a new unseen dimension of space. * He suggested that the extra dimension somehow might be distinguished from the three familiar infinite ones, though he didn’t specify how. Kaluza’s goal with this extra dimension was to unify the forces of gravity and electromagnetism. Although the details of that failed unification attempt are irrelevant here, the extra dimension that he had so brazenly introduced is very relevant indeed.
    Kaluza wrote his paper in 1919. Einstein, who was the referee evaluating it for publication in a scientific journal, wavered about the merits of the idea. Einstein delayed the publication of Kaluza’s paper for two years, but eventually acknowledged its originality. Yet Einstein still wanted to know what this dimension was. Where was it and why was it different? How far did it extend?
    These are the obvious questions to ask. They might be some of the very same questions that are bothering you. No one responded to Einstein until 1926, when the Swedish mathematician Oskar Klein addressed his questions. Klein proposed that the extra dimension would be curled up in the form of a circle, and that it would be extremely small, just 10 -33 cm, † one tenth of a millionth of a trillionthof a trillionth of a centimeter. This tiny rolled-up dimension would be everywhere: each point in space would have its own minuscule circle, 10 -33 cm in size.
    This small quantity represents the Planck length, a quantity that will be relevant later when we discuss gravity in more detail. Klein picked the Planck length because it is the only length that could naturally appear in a quantum theory of gravity, and gravity is connected to the shape of space. For now, all you need to know about the Planck length is that it is extraordinarily, unfathomably small—far smaller than anything we would ever have a chance of detecting. It is about twenty-four orders of magnitude † smaller than an atom and nineteen orders of magnitude smaller than a proton. It’s easy to overlook anything as tiny as that.
    There are many examples in daily life of objects whose extent in one of the three familiar dimensions is too small to be noticed. The paint on a wall, or a clothesline viewed from far away, are examples of things that seem to extend in fewer than three dimensions. We overlook the paint’s depth and the clothesline’s thickness. To a casual observer, the paint looks as if