Warped Passages - Lisa Randall [17]
But Athena’s cozy universe did not last long. Slipping through a gap in OneDLand, she returned to the dreamworld’s rabbit hole, which had an elevator that could take her to hypothetical, other-dimensional universes. Almost immediately, the Rabbit announced, “Next stop: TwoDLand—a two-dimensional world.” Athena didn’t think “TwoDLand” a very nice name, but she cautiously entered all the same.
Athena needn’t have been so hesitant. Almost everything in TwoDLand looked the same as in OneDLand. She did notice one difference—a vial labeled “Drink me.” Bored with one dimension, Athena promptly obeyed. She quickly shrank to a tiny size, and as she became smaller, a second dimension came into view. This second dimension was not very big—it was wrapped around in a fairly small circle. Her surroundings now resembled the surface of an extremely long tube. A Dodo was racing around this circular dimension, but he wanted to stop. So he kindly offered Athena, who looked rather hungry, some cake.
When Athena ate a morsel of the Dodo’s dreamcake, she started to grow. After only a few bites (she was quite sure of this, as she was still rather hungry), the cake very nearly disappeared; all that remained was a very tiny crumb. At least Athena thought there was a crumb, but she could see it only when she squinted very hard. And the cake wasn’t the only thing that had vanished from view: when Athena returned to her usual size, the entire second dimension had disappeared.
She thought to herself, “TwoDLand is very odd indeed. I’d best be getting home.” Her return journey was not without further adventures, but those will be kept for another time.
Even if we don’t know why three spatial dimensions are special, we can ask how. How is it possible that the universe could appear to have only three dimensions of space if the fundamental underlying spacetime contains more? If Athena is in a two-dimensional world, why does she sometimes see only one? If string theory is the correct description of nature, and there are nine dimensions of space (plus one of time), what has become of the missing six spatial dimensions? Why aren’t they visible? Do they have any discernible impact on the world we see?
The last three questions are central to this book. However, the first order of business is to determine whether there is any way in which the evidence of extra dimensions can be hidden so that Athena’s two-dimensional world would appear as one-dimensional, or a universe with extra dimensions would appear to have the three-spatial-dimensional structure we observe around us. If we’re to accept the idea of a world with extra dimensions, whatever theory they come from, there must be a good explanation for why we have not yet detected even the slightest trace of their existence.
This chapter is about extremely small compactified, or rolled-up, dimensions. They don’t extend for ever, like the three familiar dimensions; instead, they 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