Warped Passages - Lisa Randall [26]
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. Because dimensions that disappear into nothing are potentially dangerous—you wouldn’t want pieces of the universe to fall off the ends—there must be boundaries for the finite dimensions that tell them where and how to end. The question is, what happens to particles and energy when they reach these boundaries?
The answer is that they encounter a brane. In a higher-dimensional world, branes would be the boundaries of the full higher-dimensional space, known as the bulk. Unlike a brane, the bulk extends in all directions. The bulk spans every dimension, both on and off the brane (see Figure 25). The bulk is therefore “bulky,” whereas, in comparison, the brane is flat (in some dimensions), like a pancake. If branes bordered the bulk in certain directions, some of the bulk’s dimensions would be parallel to the brane, while other dimensions would lead off it. If the brane is the boundary, the dimensions off the brane would extend only to one side.
Figure 25. A brane is a lower-dimensional surface with directions along it and directions that lead away from it, into the higher-dimensional bulk.
To understand the nature of finite dimensions that end on branes, let us consider a very long thin pipe. Within the pipe there are three dimensions: one long and two short. To make the analogy to flat branes most straightforward, let’s imagine that our pipe has a square cross-section. An infinitely long pipe of this type would have four infinitely long straight walls. If the pipe were a universe in its own right, it would be one with three dimensions, two of which are bounded on either side by walls and one that extends infinitely far.
We know that a long thin pipe when viewed from afar (or with insufficient resolution) looks one-dimensional, much like the garden hose of the previous chapter. But we can also ask, as