Free Warped Passages by Lisa Randall Page A

mechanisms for confining particles on branelike surfaces. But string theory branes were the first known type of brane that could trap forces as well as particles, and we’ll soon see that is part of what makes them so interesting. Like Ike stuck on a two-dimensional road in three-dimensional space, particles and forces can be trapped on lower-dimensional surfaces called branes, even if the universe has many other dimensions to explore. If string theory accurately describes the world in which we live, physicists have no choice but to acknowledge the potential existence of such branes.
    The world of branes is an exciting new landscape that has revolutionized our understanding of gravity, particle physics, and cosmology. Branes might really exist in the cosmos, and there is no good reason that we couldn’t be living on one. Branes might even play an important role in determining the physical properties of our universe and ultimately explain observable phenomena. If they do, branes and extra dimensions will be here to stay.
    Branes as Slices
    In Chapter 1 we looked at one way of thinking about the two-dimensional world of Flatland: as a two-dimensional slice of a three-dimensional space. In Abbott’s novel, the character A. Square took a journey beyond two-dimensional Flatland, into the third dimension, and recognized that Flatland was a mere slice of the bigger three-dimensional world.
    Upon his return, A. Square suggested—logically enough—that the three-dimensional world he had seen might also be a mere slice: a three-dimensional slice of an even higher-dimensional space. By “slice,” of course, I don’t mean merely a paper-thin, two-dimensional membrane, but the logical extension of such a thing—a generalizedmembrane, if you like. You might think of the three-dimensional slices that A. Square suggested as three-dimensional chunks in four-dimensional space.
    But his three-dimensional guide promptly dismissed A. Square’s speculation about three-dimensional slices. Like almost everyone we know, this unimaginative inhabitant of three dimensions believed in only the three dimensions of space he could see; he couldn’t even contemplate a fourth.
    Branes have introduced mathematical notions into physics that are similar to those described in Flatland over a century ago. Physicists have now returned to the idea that the three-dimensional world that surrounds us could be a three-dimensional slice of a higher-dimensional world. A brane is a distinct region of spacetime that extends through only a (possibly multidimensional) slice of space. The word “membrane” motivated the choice of the word “brane” because membranes, like branes, are layers that either surround or run through a substance. Some branes are “slices” inside the space, but others are “slices” that bound space, like slices of bread in a sandwich.
    Either way, a brane is a domain that has fewer dimensions than the full higher-dimensional space that surrounds or borders it. 5 Note that membranes have two dimensions, but branes can have any number of dimensions. Although the branes that will most interest us have three spatial dimensions, the word “brane” refers to all “slices” of this sort; some branes have three spatial dimensions, but other branes have more (or fewer). 6 We’ll use 3-branes to refer to branes with three dimensions, 4-branes to refer to those with four, and so on.
    Boundary Branes and Embedded Branes
    In the previous chapter I explained why we might not see extra dimensions. They could be curled up into sizes so small that evidence of their existence never would appear. The key point was that the extra dimensions would be small. None of the reasons for the invisibility of dimensions relied on the fact that extra dimensions were curled up.
    This suggests an alternative possibility: perhaps dimensions are not rolled up, but simply terminate within a finite distance. Becausedimensions that disappear into nothing are