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are massless.) Excessive numbers of massless particles are a particularly awkward feature of any proposed theory since there are tight limits on such particles coming from both accelerator data and cosmological observations. For string theory to be viable it is imperative that these particles acquire mass. In recent years, various discoveries have revealed ways in which this might happen, having to do with fluxes that can thread through holes in the extra-dimensional Calabi-Yau shapes. I will discuss aspects of these developments in Chapter 5.

14. It is not impossible for experiments to provide evidence that would strongly disfavor string theory. The structure of string theory ensures that certain basic principles should be respected by all physical phenomena. Among these are unitarity (the sum of all probabilities of all possible outcomes in a given experiment must be 1) and local Lorentz invariance (in a small enough domain the laws of special relativity hold), as well as more technical features such as analyticity and crossing symmetry (the result of particle collisions must depend on the particles’ momentum in a manner that respects a particular collection of mathematical criteria). Should evidence be found—perhaps at the Large Hadron Collider—that any of these principles are violated, it would be a challenge to reconcile those data with string theory. (It would also be a challenge to reconcile those data with the standard model of particle physics, which incorporates these principles too, but the underlying assumption is that the standard model must give way to some kind of new physics at a high enough energy scale since the theory does not incorporate gravity. Data conflicting with any of the principles enumerated would argue that the new physics is not string theory.)

15. It is common to speak of the center of a black hole as if it were a position in space. But it’s not. It is a moment in time. When crossing the event horizon of a black hole, time and space (the radial direction) interchange roles. If you fall into a black hole, for example, your radial motion represents progress through time. You are thus pulled toward the black hole’s center in the same way you are pulled to the next moment in time. The center of the black hole is, in this sense, akin to a last moment in time.

16. For many reasons, entropy is a key concept in physics. In the case discussed, entropy is being used as a diagnostic tool to determine if string theory is leaving out any essential physics in its description of black holes. If it was, the black hole disorder that the string mathematics is being used to calculate would be inaccurate. The fact that the answer agrees exactly with what Bekenstein and Hawking found using very different considerations is a sign that string theory has successfully captured the fundamental physical description. This is a very encouraging result. For more details, see The Elegant Universe, Chapter 13.

17. The first hint of this pairing between Calabi-Yau shapes came from the work of Lance Dixon, as well as independently from Wolfgang Lerche, Nicholas Warner, and Cumrun Vafa. My work with Ronen Plesser found a method for producing the first concrete examples of such pairs, which we named mirror pairs, and the relationship between them mirror symmetry. Plesser and I also showed that difficult calculations on one member of a mirror pair, involving seemingly impenetrable details such as the number of spheres that can be packed into the shape, could be translated into far more manageable calculations on the mirror shape. This result was picked up by Philip Candelas, Xenia de la Ossa, Paul Green, and Linda Parkes and put into action—they developed techniques for explicitly evaluating the equality Plesser and I had established between the “difficult” and “easy” formulas. Using the easy formula, they then extracted information about its difficult partner, including the numbers associated with the sphere packing given in the text. In the years since, mirror symmetry has become its own field of research, with a great