Warped Passages - Lisa Randall [221]
How can a four-dimensional and a five-(or ten-)dimensional theory have the same physical implications? What is the analog of an object traveling through the fifth dimension, for example? The answer is that an object moving through the fifth dimension would appear in the dual four-dimensional theory as an object that grows or shrinks. This is just like Athena’s shadow on the Gravitybrane, which grew as she moved away from the Gravitybrane across the fifth dimension. Furthermore, objects moving past each other along the fifth dimension correspond to objects that grow and shrink and overlap in four dimensions.
Once you introduce branes, the consequences of the duality are even stranger. For example, five-dimensional anti de Sitter space with gravity but without branes is equivalent to a four-dimensional theory without gravity. But once you include a brane in the five-dimensional theory, as Raman and I did, the equivalent four-dimensional theory suddenly contains gravity.
Does this duality mean that I was cheating when I said that the warped geometries were higher-dimensional theories? Absolutely not. The duality is intriguing, but it doesn’t really change anything I’ve told you. Even if someone finds the precise dual four-dimensional theory, such a theory will be extremely difficult to study. It has to contain an enormous number of particles and such extremely strong interactions that perturbation theory (see Chapter 15) wouldn’t apply.
Theories in which objects strongly interact are almost always impossible to interpret without an alternate, weakly interacting description. And in this case, that tractable description is the five-dimensional theory. Only the five-dimensional theory has a simple enough formulation to use for computation, so it makes sense to think of the theory in five-dimensional terms. Nonetheless, even if the five-dimensional theory is more tractable, duality still makes me wonder what the word “dimensions” really means. We know that the number of dimensions should be the number of quantities you need to specify the location of an object. But are we always sure we know which quantities to count?
II. T-duality
Another reason to question the meaning of dimensions is an equivalence between two superficially different geometries that is known as T-duality. Even before string theorists discovered any of the dualities I’ve discussed, they discovered T-duality, which exchanges a space with a tiny rolled-up dimension for another space with a huge rolled-up dimension.39 Odd as it may seem, in string theory, extremely small and extremely large rolled-up dimensions yield the same physical consequences. A minuscule tiny volume of rolled-up space has the same physical consequences as an extremely large one.
T-duality applies in string theory with curled-up dimensions because there are two different types of closed string in spacetime compactified on a circle, and these two strings get interchanged when a space with a tiny rolled-up dimension is exchanged for a space with a large one. The first type of closed string oscillates up and down as it circles the closed dimension, similar to the behavior of the Kaluza-Klein particles we looked at in Chapter 18. The other type wraps around the curled-up dimension. It can do so once, twice, or any number of times. And T-duality operations, which interchange a small rolled-up space for a large one, exchange these two types of string.
In fact, T-duality was the first clue that branes had to exist: without them, open strings wouldn’t have had analogs in the dual theory. But if T-duality does apply and a minuscule rolled-up dimension yields the same physical consequences as an enormous rolled-up dimension, it would mean that, once again, our notion of “dimension” is inadequate.
That is because if you imagine making the radius of one rolled-up dimension infinitely large, the T-dual rolled-up dimension would be a circle of zero size—there would be no circle at all. That