The Elegant Universe - Brian Greene [219]
14. We should note, though, that even in the peninsular regions there are some exotic ways in which branes can have an effect on familiar physics. For example, it has been suggested that our three extended spatial dimensions might themselves be a three-brane that is large and unfurled. If so, as we go about our daily business we would be gliding through the interior of a three-dimensional membrane. Investigations of such possibilities are now being undertaken.
15. Interview with Edward Witten, May 11, 1998.
Chapter 13
1. The expert reader will recognize that under mirror symmetry, a collapsing three-dimensional sphere on one Calabi-Yau space gets mapped to a collapsing two-dimensional sphere on the mirror Calabi-Yau space—apparently putting us back in the situation of flops discussed in Chapter 11. The difference, however, is that a mirror rephrasing of this sort results in the antisymmetric tensor field Bµv—the real part of the complexified Kähler form on the mirror Calabi-Yau space—vanishing, and this is a far more drastic sort of singularity than that discussed in Chapter 11.
2. More precisely, these are examples of extremal black holes: black holes that have the minimum mass consistent with the force charges they carry, just like the BPS states in Chapter 12. Similar black holes will also play a pivotal role in the following discussion on black hole entropy.
3. The radiation emitted from a black hole should be just like that emitted from a hot oven—the very problem, discussed at the outset of Chapter 4, that played such a pivotal role in the development of quantum mechanics.
4. It turns out that because the black holes involved in space-tearing conifold transitions are extremal, they do not Hawking radiate, regardless of how light they become.
5. Stephen Hawking, lecture at Amsterdam Symposium on Gravity, Black Holes, and Strings, June 21, 1996.
6. In their initial calculation, Strominger and Vafa found that the mathematics was made easier by working with five—not four—extended spacetime dimensions. Surprisingly, after completing their calculation of the entropy of such a five-dimensional black hole they realized that no theoretician had as yet constructed such hypothetical extremal black holes in the setting of five-dimensional general relativity. Since only by comparing their answer to the area of the event horizon of such a hypothetical black hole could they confirm their results, Strominger and Vafa then set out to mathematically construct such a five-dimensional black hole. They succeeded. It was then a simple matter to show that the microscopic string theory calculation of the entropy was in agreement with what Hawking would have predicted based on the area of the black hole's event horizon. But it is interesting to realize that because the black hole solution was found later, Strominger and Vafa did not know the answer they were shooting for while undertaking their entropy calculation. Since their work, numerous researchers, led most notably by Princeton physicist Curtis Callan, have succeeded in extending the entropy calculations to the more familiar setting of four extended spacetime dimensions, and all are in agreement with Hawking's predictions.
7. Interview with Sheldon Glashow, December 29, 1997.
8. Laplace, Philosophical Essay on Probabilities, trans. Andrew I. Dale (New York: Springer-Verlag, 1995).
9. Stephen Hawking, in Hawking and Roger Penrose, The Nature of Space and Time (Princeton: Princeton University Press, 1995), p. 41.
10. Stephen Hawking, lecture at the Amsterdam Symposium on Gravity, Black Holes, and Strings, June 21, 1997.
11. Interview