Warped Passages - Lisa Randall [235]
36. The metric in the warped geometry is ds2 = e-kr(dx2 + dy2 + dz2 + c2dt2) + dr2, where r is the coordinate of the fifth dimension. That tells us that at any fixed location in the fifth dimension, which corresponds to fixed r, spacetime is completely flat. However, the overall r-dependent factor tells us that how we measure size changes according to the position of an object in the fifth dimension. The exponential falloff of the coefficient, which is the warp factor, is the reason that the graviton’s probability function falls exponentially, and is also why we need to rescale mass, energy, and size to make a single, four-dimensional effective theory.
37. Because space is not flat, the extra-dimensional volume that enters when we calculate MPl in four dimensions is not simply MPl3R, as it would be when space is flat. Instead, the value of MPl depends on the curvature. If the metric has the form ds2 = e-kr(dx2 + dy2 + dz2 − c2dt2) + dr2, where r is the coordinate of the fifth dimension, then, roughly, MPl2 = M3/k. In other words, the size of the space is largely irrelevant. This makes sense because the curvature of space—not the extra dimension’s size—determines how the field lines spread in the extra dimension and hence the strength of four-dimensional gravity. In fact, there is small dependence on R: the real formula is MPl2 = M3/k(1 − e-kR), but when kR is big, the exponential term is largely irrelevant and can be neglected.
38. The warp factor in the locally localized gravity model that Andreas Karch and I developed is the sum of a decreasing exponential function (like the warped geometries we have already considered) and an increasing exponential function. It is proportional to cosh(kc − kr), where k is related to the bulk energy and c is related to the brane energy. Like the localized gravity warp factor we have already considered, this warp factor falls exponentially as you leave the brane. But unlike the previous case, the warp factor turns around and then exponentially increases. The four-dimensional graviton is localized in the region between the brane and this “turnaround” point. Beyond that distance, four-dimensional gravity no longer applies.
39. Under T-duality, the compactification radius, r, is interchanged with its inverse, 1/r (with distances measured in units of the string length).
40. The physicists Csaba Csaki, Joshua Erlich, and Christophe Grojean have, however, made the interesting observation that the speed of light and the speed of gravity can be different (the speed of gravity can actually be faster) if there is an asymmetrically warped spacetime in which the scalings of time and spatial coordinates along a fifth dimension are different from each other.
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