Warped Passages (84 page)
†
Joseph Polchinski,
String Theory, Vol. 1: An Introduction to the Bosonic String
(Cambridge: Cambridge University Press, 1998).
*
Although we can do perturbation theory when the coupling is very weak, or when there is a weakly coupled dual description of a theory with strong interactions, we have no way of using perturbation theory when the interaction strength is in the middle—that is, about 1. That means that even when there is a dual description, we don’t have a complete solution to the theory.
*
Really it is a bound state of D
0
branes.
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Francis Bacon,
On Scientific Inquiry
.
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John Ellis, Costas Kounnas, and Dmitri Nanopoulos had also considered related ideas in string theory earlier on.
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K. Square in the story. KK particles are also known as Kaluza-Klein modes, where “modes” refers to their quantized momenta.
*
This is our usual counting of spacetime dimensions. Our previous discussion of Flatland in Chapter 1 preceded relativity, so we only counted spacial dimensions there.
*
Remember, we have assumed no branes; this limit will change in the following chapters.
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Physicists post their papers on a website that begins with “xxx”: check out xxx.lanl.gov. Internet filters have occasionally forbidden access to this site as well.
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For brevity, I’ll refer to them collectively as “ADD.”
*
Nima Arkani-Hamed, Savas Dimopoulos, Gia Dvali, “The hierarchy problem and new dimensions at a millimeter,”
Physics Letters
B, vol. 429, pp. 263–72 (1998).
*
Remember the Planck scale
length
is tiny, but the Planck scale
mass
(or
energy
) is enormous.
*
If they are flat (see Chapter 22).
*
As we saw in Chapter 18, extra dimensions can be uniform, large, and flat. The Rabbit is skeptical about this idea.
†
This counting includes a dimension of time.
*
Tap water, bottled water, water with gas, water without, etc.
*
I’ll sometimes use “three-plus-one” instead of “four” when I want to emphasize the distinction between space and time.
†
Really, all the slices have the same geometry; in this case, the slices are all flat.
*
Remember that the fifth dimension is the fifth dimension of spacetime and the hypothetical fourth dimension of space.
*
The units in which distance is measured are determined by the energy on the brane, which would be determined by the Planck scale mass.
†
This number is in units of the curvature, which is in turn determined by the energy on the brane and in the bulk.
*
The names Planckbrane and TeV-brane or Weakbrane are the terms commonly used in the physics literature. Gravitybrane will be Branesville in the story in the next chapter. The name Weakbrane refers to the fact that most particles confined to this brane are expected to have a mass about the size of the weak scale mass.
*
Kaustubh Agashe, Roberto Contino, Michael J. May, Alex Pomarol, and Raman Sundrum are among the physicists who have studied detailed models of what might be present.
*
This title borrows from Martin Gardner’s delightful
Annotated Alice
, in which he explains the wordplay, math riddles, and references in Lewis Carroll’s
Alice in Wonderland
and
Through the Looking Glass
.
†
The brane itself is large and flat and has only three spatial dimensions. Only gravity makes contact with the additional dimension. Remember that the five-dimensional space has four spatial dimensions (and one of time), whereas the brane has three spatial dimensions. I’ll still call time the fourth dimension, and I’ll call the additional dimension the fifth.
‡
Branesville is the Gravitybrane.
*
The Fat Cat, unlike Branesville residents, is not confined to the brane.
†
Everything is bigger and lighter near the Weakbrane. Athena’s shadow over Branesville grew as she got closer to the Weakbrane and further away from the Gravitybrane.
*
The fifth dimension does not have to be very big in order to solve the hierarchy problem.
†
Gravity is feeble on the Weakbrane, where the graviton’s probability function is so small.
*
On the Gravitybrane, gravity is no weaker than the other forces.
†
The petulant graviton is complaining that on the Weakbrane, gravity is much weaker than the electromagnetic, weak, and strong forces. Gravity would be much stronger (and would have a strength closer to that of the other forces) closer to the Gravitybrane.
‡
Things are bigger and time is slower on the Weakbrane. The rabbit’s laxness is accounted for by rescaling time.
*
The geometry of this chapter is warped, as in the previous ones, but now there is only a single brane—the Gravitybrane. Although this means that there is an infinite fifth dimension, this chapter will show why this is perfectly fine with the warped spacetime.
*
We consider a straight sprinkler, instead of the circular one that we considered before, because it is easier to generalize to the warped scenario.
*
A real-life analogy of this sort would be the Colorado River, where dams and irrigation ensure that water is delivered to the Southwestern United States, but only a small amount of water remains in the river when it reaches Mexico. Putting a dam near the Gulf of California (which would be like putting another brane far from the Gravitybrane) wouldn’t affect the amount of water that Las Vegas receives.
*
The curled-up space is still mathematically “flat.” That is because you can unroll the dimension to something you would recognize as flat; that is not true of a sphere, for example.
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They include Juan Garcia-Bellido, Andrew Chamblin, Roberto Emparan, Ruth Gregory, Stephen Hawking, Gary T. Horowitz, Nemanja Kaloper, Robert C. Myers, Harvey S. Reall, Hisa-aki Shinkai, Tetsuya Shiromizu, and Toby Wiseman.
*
This model is also known as “KR,” after the initials of our last names.
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Manifolds can have different numbers of holes; for example, a sphere has no holes, whereas a torus—a donut-like shape—has one.
*
T. Banks, W. Fischler, S.H. Shenker, and L. Susskind, “M theory as a matrix model: a conjecture,”
Physical Review
D, vol. 55, pp. 5112–28 (1997).
*
The quotes are from K.C. Cole’s article, “Time, space obsolete in new view of universe,”
Los Angeles Times
, November 16, 1999.
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