The Elegant Universe - Brian Greene [220]
12. Interview with Cumrun Vafa, January 12, 1998.
13. Stephen Hawking, lecture at the Amsterdam Symposium on Gravity, Black Holes, and Strings, June 21, 1997.
14. This issue also has some bearing on the information-loss question, as some physicists have speculated over the years that there might be a central "nugget" embedded in the depths of a black hole that stores all of the information carried by matter that gets trapped within the hole's horizon.
15. In fact, the space-tearing conifold transitions discussed in this chapter involve black holes and hence might seem to be tied up with the question of their singularities. But recall that the conifold tear occurs just as the black hole has shed all its mass, and is therefore not directly related to questions concerning black hole singularities.
Chapter 14
1. More precisely, the universe should be filled with photons conforming to the radiation thermally emitted by a perfectly absorbent body—a "black-body" in the language of thermodynamics—with the stated temperature range. This is the same radiation spectrum emitted quantum mechanically by black holes, as explained by Hawking, and by a hot oven, as explained by Planck.
2. The discussion conveys the spirit of the issues involved although we are glossing over some subtle features having to do with the motion of light in an expanding universe that affect the detailed numerics. In particular, although special relativity declares that nothing can travel faster than the speed of light, this does not preclude two photons carried along on the expanding spatial fabric from receeding from one another at a speed exceeding that of light. For example, at the time the universe first became transparent, about 300,000 years ATB, locations in the heavens that were about 900,000 light-years apart would have been able to have influenced each other, even though the distance between them exceeds 300,000 light-years. The extra factor of three comes from the expansion of the spatial fabric. This means that as we run the cosmic film backward in time, by the time we get to 300,000 years ATB, two points in the heavens need only be less than 900,000 light-years apart to have had a chance to influence each other's temperature. These detailed numerics do not change the qualitative features of the issues discussed.
3. For a detailed and lively discussion of the discovery of the inflationary cosmological model and the problems it resolves, see Alan Guth, The Inflationary Universe (Reading, Mass: Addison-Wesley, 1997).
4. For the mathematically inclined reader, we note that the idea underlying this conclusion is the following: If the sum of the spacetime dimensions of the paths swept out by each of two objects is greater than or equal to the spacetime dimension of the arena through which they are moving then they will generically intersect. For instance, point particles sweep out one-dimensional spacetime paths—the sum of the spacetime dimensions for two such particle paths is therefore two. The spacetime dimension of Lineland is also two, and hence their paths will generally intersect (assuming their velocities have not been finely tuned to be exactly equal). Similarly, strings sweep out two-dimensional spacetime paths (their world-sheets); for two strings the sum in question is therefore four. This means that strings moving in four spacetime dimensions (three space and one time) will generally intersect.
5. With the discovery of M-theory and the recognition of an eleventh dimension, string theorists have begun studying ways of curling up all seven extra dimensions in a manner that puts them all on more or less equal footing. The possible choices for such seven-dimensional manifolds are known as Joyce manifolds, after Domenic Joyce of Oxford University, who is credited with finding the first techniques for their mathematical construction.
6. Interview with Cumrun Vafa, January 12, 1998.
7. The expert reader will note that our description is taking place in the so-called string frame of reference,