Warped Passages - Lisa Randall [234]
27. They can also extend in zero dimensions, in which case they are new kinds of particle called Do-branes, or in one dimension, in which case they are new types of string called D1-branes.
28. Branes don’t necessarily interact via ordinary charges. They interact via a higher-dimensional generalization of charges.
29. The symmetry actually rotates branes into each other, but this is beyond the technical reach of this book (and would make Igor’s head spin).
30. Ordinarily, the gaugino masses fall in a ratio of about 1:3:30, where the photino is the lightest, winos are next (though the zino might be a little heavier or lighter than the winos), and the gluinos are heaviest. In sequestered models the ratio is 1:2:8, where the winos are the lightest, the photino is heavier, and the gluino is again the heaviest.
31. The wavefunctions of the Kaluza-Klein modes are the modes that occur in the generalized Fourier decomposition of the higher-dimensional wavefunction.
32. This also assumes that there are no singularities in the spacetime geometry—that is, no place where the space shrinks to zero size.
33. D. Cremades, S. Franco, L. Ibanez, F. Marchesano, R. Rabadan, and A. Uranga also suggested an interesting alternative. Their idea is that particles are not confined on an individual brane, but are instead confined to the intersections of multiple branes. As with separated parallel branes, strings extending between branes will generally be heavy. But light or massless particles arise from zero-length strings, which in this case would be confined to the region where the branes intersect.
34. We can also show this in a slightly different way with a more mathematical argument. When there are curled-up dimensions, the force lines emanating from a massive object behave according to the gravitational law of the higher-dimensional theory at short distances and according to four-dimensional gravity at long distances. The only way to reconcile the two force laws and switch smoothly from one to the other is by noting that at about the distance corresponding to the extra dimensions’ sizes, the force lines spread as if there were only four dimensions, but with a reduced strength because of the extra volume of the curled-up space. Beyond the size of the extra dimensions, gravity behaves four-dimensionally but with its strength suppressed by the spreading out over the extra-dimensional volume.
Newton’s law of gravitation says that when there are three spatial dimensions, the force is proportional to 1/MPl2 × 1/r2. If there are n additional dimensions, the force law would be 1/Mn + 2 × 1/rn + 2, where M sets the strength of higher-dimensional gravity, similarly to the way in which MPl sets the strength of four-dimensional gravity. Notice that the higher-dimensional force law varies more quickly with r since the force lines would spread over a hypersphere whose surface would have n + 2 dimensions (as opposed to the two-dimensional surface of a sphere that gives rise to the force law of three-dimensional space). However, when the extra-dimensional volume is finite and the n extra dimensions have size R, the force law will be 1/Mn + 2 × 1/Rn x 1/r2 when r is greater than R, and the force lines can no longer spread in the extra dimensions. This is the form of a three-spatial-dimensional force law if we make the identification MPl2 = Mn + 2 Rn. Since Rn is the volume of the higher-dimensional space, we find that the strength of gravity decreases with volume, or equivalently (because gravity’s strength is weaker when the Planck scale energy is bigger), the Planck scale energy is big if the volume is big.
35. A flat metric with three spatial dimensions is ds2 = dx2 + dy2 + dz2 + c2dt2. Because there are no spatial or time-dependent coefficients, measurements are independent of where you are or which direction you point in; that is to say, spacetime is completely flat. All three spatial coordinates as well as the time coordinate (up to the minus sign that always singles time out) are treated the same; that is, the coefficients of the terms in the