Warped Passages - Lisa Randall [20]
The two-dimensional world in which the dreaming Athena found herself was very much like the garden-hose universe. Because Athena had the opportunities to be both big and small relative to TwoDLand’s width, she could observe this universe from both the perspective of someone bigger and that of someone smaller than its second dimension. To the big Athena, TwoDLand and OneDLand appeared the same in every respect. Only the small Athena could tell the difference. Similarly, in the garden-hose universe a being would be ignorant of an additional spatial dimension if it were too tiny for it to see.
Let’s now return to the Kaluza-Klein universe, which has the three spatial dimensions we know about, supplemented by an extra one that’s unseen. We can again use Figure 16 to think about this situation. Ideally, I would draw four spatial dimensions, but unfortunately that’s not possible (even a pop-up book wouldn’t suffice). However, since the three infinite dimensions that constitute our space are all qualitatively the same, I really need only draw just one representative dimension. That leaves me free to use the other dimension to represent the unseen extra dimension. The other dimension shown here is the one that’s curled up—the one that’s fundamentally different from the other three.
Just as with our two-dimensional garden-hose universe, a four-dimensional Kaluza-Klein universe with a single tiny, rolled-up dimension would appear to us to have one dimension fewer than the four it actually has. Because we wouldn’t know about the additional spatial dimension unless we could detect evidence of structure on its minute scale, the Kaluza-Klein universe would look three-dimensional. Rolled-up, or compactified, extra dimensions will never be detected if they are sufficiently tiny. Later on, we’ll investigate just how tiny, but for now, rest assured that the Planck length is well below the threshold of detectability.
In life, and in physics, we only register those details that actually matter to us. If you cannot observe detailed structure, you might as well pretend it isn’t there. In physics, this disregard of local detail is embodied in the effective theory idea of the previous chapter. In an effective theory, all that matters are the things that you can actually perceive. In the example above, we would use a three-dimensional effective theory where the information about extra dimensions is suppressed.
Although the curled-up dimension of the Kaluza-Klein universe is not far away, it’s so small that any variation within it is imperceptible. Just as differences among New Yorkers don’t really matter to people outside, the structure in the extra dimensions of the universe is irrelevant when its details vary on such minuscule a scale. Even if fundamentally there turn out to be many more dimensions than we acknowledge in our daily lives, everything we see can still be described in terms of only the dimensions we observe. Extremely small extra dimensions change nothing about the way we view the world, or even about how we do most physics calculations. Even if additional dimensions exist, if we are incapable of seeing or experiencing them, we can ignore them and still correctly describe what we see. Later on we’ll see modifications to this simple picture for which this won’t always be true, but those will involve additional assumptions.
We can understand one further important point about a rolled-up dimension from Figure 17, which illustrates the hose, or universe with one