The Fabric of the Cosmos: Space, Time, and the Texture of Reality (73 page)

7. It is somewhat of a misnomer to speak of the "center" of a black hole as if it were a place in space. The reason, roughly speaking, is that when one crosses a black hole's event horizon—its outer edge—the roles of space and time are interchanged. In fact, just as you can't resist going from one second to the next in time, so you can't resist being pulled to the black hole's "center" once you've crossed the event horizon. It turns out that this analogy between heading forward in time and heading toward a black hole's center is strongly motivated by the mathematical description of black holes. Thus, rather than thinking of the black hole's center as a location in space, it is better to think of it as a location in time. Furthermore, since you can't go beyond the black hole's center, you might be tempted to think of it as a location in spacetime where time comes to an end. This may well be true. But since the standard general relativity equations break down under such extremes of huge mass density, our ability to make definite statements of this sort is compromised. Clearly, this suggests that if we had equations that don't break down deep inside a black hole, we might gain important insights into the nature of time. That is one of the goals of superstring theory.

8. As in earlier chapters, by "observable universe" I mean that part of the universe with which we could have had, at least in principle, communication during the time since the bang. In a universe that is infinite in spatial extent, as discussed in Chapter 8, all of space does
not
shrink to a point at the moment of the bang. Certainly, everything in the observable part of the universe will be squeezed into an ever smaller space as we head back to the beginning, but, although hard to picture, there are things—infinitely far away—that will forever remain separate from us, even as the density of matter and energy grows ever higher.

9. Leonard Susskind, in "The Elegant Universe,"
NOVA,
three-hour PBS series first aired October 28 and November 4, 2003.

10. Indeed, the difficulty of designing experimental tests for superstring theory has been a crucial stumbling block, one that has substantially hindered the theory's acceptance. However, as we will see in later chapters, there has been much progress in this direction; string theorists have high hopes that upcoming accelerator and space-based experiments will provide at least circumstantial evidence in support of the theory, and with luck, maybe even more.

11. Although I haven't covered it explicitly in the text, note that every known particle has an
antiparticle—
a particle with the same mass but opposite force charges (like the opposite sign of electric charge). The electron's antiparticle is the positron; the up-quark's antiparticle is, not surprisingly, the anti-up-quark; and so on.

12. As we will see in Chapter 13, recent work in string theory has suggested that strings may be much larger than the Planck length, and this has a number of potentially critical implications—including the possibility of making the theory experimentally testable.

13. The existence of atoms was initially argued through indirect means (as an explanation of the particular ratios in which various chemical substances would combine, and later, through Brownian motion); the existence of the first black holes was confirmed (to many physicists' satisfaction) by seeing their effect on gas that falls toward them from nearby stars, instead of "seeing" them directly.

14. Since even a placidly vibrating string has
some
amount of energy, you might wonder how it's possible for a string vibrational pattern to yield a massless particle. The answer, once again, has to do with quantum uncertainty. No matter how placid a string is, quantum uncertainty implies that it has a minimal amount of jitter and jiggle. And, through the weirdness of quantum mechanics, these uncertainty-induced jitters have
negative
energy. When this is combined with the positive energy from the most gentle of ordinary string vibrations, the total mass/energy is zero.

15. For the mathematically inclined reader, the more precise statement is that the
square
of the masses of string vibrational modes are given by integer multiples of the square of the Planck mass. Even more precisely (and of relevance to recent developments covered in Chapter 13), the square of these masses are integer multiples of the
string scale
(which is proportional to the inverse square of the string length). In conventional formulations of string theory, the string scale and the Planck mass are close, which is why I've simplified the main text and only introduced the Planck mass. However, in Chapter 13 we will consider situations in which the string scale can be different from the Planck mass.

16. It's not too hard to understand, in rough terms, how the Planck length crept into Klein's analysis. General relativity and quantum mechanics invoke three fundamental constants of nature:
c
(the velocity of light),
G
(the basic strength of the gravitational force) and (Planck's constant describing the size of quantum effects). These three constants can be combined to produce a quantity with units of length: ( G/c
³
)
^1/2
, which, by definition, is the Planck length. After substituting the numerical values of the three constants, one finds the Planck length to be about 1.616 × 10
^-33
centimeters. Thus, unless a dimensionless number with value differing substantially from 1 should emerge from the theory—something that doesn't often happen in a simple, well-formulated physical theory—we expect the Planck length to be the characteristic size of lengths, such as the length of the curled-up spatial dimension. Nevertheless, do note that this does not rule out the possibility that dimensions can be larger than the Planck length, and in Chapter 13 we will see interesting recent work that has investigated this possibility vigorously.

17. Incorporating a particle with the electron's charge, and with its relatively tiny mass, proved a formidable challenge.

18. Note that the uniform symmetry requirement that we used in Chapter 8 to narrow down the shape of the universe was motivated by astronomical observations (such as those of the microwave background radiation) within the
three large dimensions.
These symmetry constraints have no bearing on the shape of the possible six tiny extra space dimensions. Figure 12.9a is based on an image created by Andrew Hanson.

19. You might wonder about whether there might not only be extra space dimensions, but also extra time dimensions. Researchers (such as Itzhak Bars at the University of Southern California) have investigated this possibility, and shown that it is at least possible to formulate theories with a second time dimension that seem to be physically reasonable. But whether this second time dimension is really on a par with the ordinary time dimension or is just a mathematical device has never been settled fully; the general feeling is more toward the latter than the former. By contrast, the most straightforward reading of string theory says that the extra space dimensions are every bit as real as the three we know about.

20. String theory experts (and those who have read
The Elegant Universe,
Chapter 12) will recognize that the more precise statement is that certain formulations of string theory (discussed in Chapter 13 of this book) admit limits involving eleven spacetime dimensions. There is still debate as to whether string theory is best thought of as fundamentally being an eleven spacetime dimensional theory, or whether the eleven dimensional formulation should be viewed as a particular limit (e.g., when the string coupling constant is taken large in the Type IIA formulation), on a par with other limits. As this distinction does not have much impact on our general-level discussion, I have chosen the former viewpoint, largely for the linguistic ease of having a fixed and uniform total number of dimensions.

Chapter 13

1. For the mathematically inclined reader: I am here referring to
conformal
symmetry—symmetry under arbitrary angle-preserving transformations on the volume in spacetime swept out by the proposed fundamental constituent. Strings sweep out two-spacetime-dimensional surfaces, and the equations of string theory are invariant under the two-dimensional conformal group, which is an
infinite
dimensional symmetry group. By contrast, in other numbers of space dimensions, associated with objects that are not themselves one-dimensional, the conformal group is finite-dimensional.

2. Many physicists contributed significantly to these developments, both by laying the groundwork and through follow-up discoveries: Michael Duff, Paul Howe, Takeo Inami, Kelley Stelle, Eric Bergshoeff, Ergin Szegin, Paul Townsend, Chris Hull, Chris Pope, John Schwarz, Ashoke Sen, Andrew Strominger, Curtis Callan, Joe Polchinski, Petr Ho ava, J. Dai, Robert Leigh, Hermann Nicolai, and Bernard deWit, among many others.

3. In fact, as explained in Chapter 12 of
The Elegant Universe,
there is an even tighter connection between the overlooked tenth spatial dimension and
p-
branes. As you increase the size of the tenth spatial dimension in, say, the type IIA formulation, one-dimensional strings stretch into two-dimensional inner-tube-like membranes. If you assume the tenth dimension is very small, as had always been implicitly done prior to these discoveries, the inner tubes look and behave like strings. As is the case for strings, the question of whether these newly found branes are indivisible or, instead, are made of yet finer constituents, remains unanswered. Researchers are open to the possibility that the ingredients so far identified in string/M-theory will not bring to a close the search for
the
elementary constituents of the universe. However, it's also possible that they will. Since much of what follows is insensitive to this issue, we'll adopt the simplest perspective and imagine that all the ingredients—strings and branes of various dimensions—are fundamental. And what of the earlier reasoning, which suggested that fundamental higher dimensional objects could not be incorporated into a physically sensible framework? Well, that reasoning was itself rooted in another quantum mechanical approximation scheme—one that is standard and fully battle tested but that, like any approximation, has limitations. Although researchers have yet to figure out all the subtleties associated with incorporating higher-dimensional objects into a quantum theory, these ingredients fit so perfectly and consistently within all five string formulations that almost everyone believes that the feared violations of basic and sacred physical principles are absent.

4. In fact, we could be living on an even higher-dimensional brane (a four-brane, a five-brane . . .) three of whose dimensions fill ordinary space, and whose other dimensions fill some of the smaller, extra dimensions the theory requires.

5. The mathematically inclined reader should note that for many years string theorists have known that closed strings respect something called T-duality (as explained further in Chapter 16, and in Chapter 10 of
The Elegant Universe
). Basically, T-duality is the statement that if an extra dimension should be in the shape of a circle, string theory is completely insensitive to whether the circle's radius is
R
or 1/
R
. The reason is that strings can move around the circle ("momentum modes") and/or wrap around the circle ("winding modes") and, under the replacement of
R
with 1/
R
, physicists have realized that the roles of these two modes simply interchange, keeping the overall physical properties of the theory unchanged. Essential to this reasoning is that the strings are closed loops, since if they are open there is no topologically stable notion of their winding around a circular dimension. So, at first blush, it seems that open and closed strings behave completely differently under T-duality. With closer inspection, and by making use of the Dirichlet boundary conditions for open strings (the "D" in D-branes), Polchinski, Dai, Leigh, as well as Ho ava, Green, and other researchers resolved this puzzle.

6. Proposals that have tried to circumvent the introduction of dark matter or dark energy have suggested that even the accepted behavior of gravity on large scales may differ from what Newton or Einstein would have thought, and in that way attempt to account for gravitational effects incompatible with solely the material we can see. As yet, these proposals are highly speculative and have little support, either experimental or theoretical.

7. The physicists who introduced this idea are S. Giddings and S. Thomas, and S. Dimopoulus and G. Landsberg.

8. Notice that the contraction phase of such a bouncing universe is not the same as the expansion phase run in reverse. Physical processes such as eggs splattering and candles melting would happen in the usual "forward" time direction during the expansion phases and would continue to do so during the subsequent contraction phase. That's why entropy would increase during both phases.

9. The expert reader will note that the cyclic model can be phrased in the language of four-dimensional effective field theory on one of the three-branes, and in this form it shares many features with more familiar scalar-field-driven inflationary models. When I say "radically new mechanism," I am referring to the conceptual description in terms of colliding branes, which in and of itself is a striking new way of thinking about cosmology.

10. Don't get confused on dimension counting. The two three-branes, together with the space interval between them, have four dimensions. Time brings it to five. That leaves six more for the Calabi-Yau space.

11. An important exception, mentioned at the end of this chapter and discussed in further detail in Chapter 14, has to do with inhomogeneities in the gravitational field, so-called primordial gravitational waves. Inflationary cosmology and the cyclic model differ in this regard, one way in which there is a chance that they may be distinguished experimentally.