Warped Passages (69 page)

We now know that gravity can behave very differently on different length scales. At very short distances, only a quantum theory of gravity such as string theory will describe gravity. On larger scales, general relativity applies admirably well, but recent observations across the universe at very big distances pose cosmological puzzles, such as what accelerates its expansion. And at longer distances still, we reach the cosmological horizon beyond which we know nothing.

One of the intriguing aspects of extra-dimensional theories is that they naturally have different consequences on different scales. Gravity in such theories exhibits behavior at distances smaller than curled-up dimensions, or where the curvature is too small to have an effect, that is different from the behavior at larger distances where dimensions might be invisible, or warping can be important. This gives us reason to believe that extra dimensions might eventually shed light on some of the mysterious features of the cosmos. If we do live in a multidimensional world, we certainly won’t be able to neglect its cosmological implications. Some research has already been done on this subject, but I’m sure many more interesting results await us.

Where do I expect physics to go from here? There are too many possibilities to enumerate. But let me describe a few intriguing obser
vations that suggest that more important theoretical surprises lie in store—ones that might come closer to resolution some time soon. These mysteries center on a question that at this point might sound shocking, namely:

What Are Dimensions, Anyway?

How can I ask such a question this late in the game? I’ve already spent much of this book discussing the meaning of dimensions and some of the potential implications of proposed extra-dimensional worlds. But now that I’ve told you what we understand about dimensions, allow me to return briefly to this question.

What does the number of dimensions really mean? We know that the number of dimensions is defined as the number of quantities that you need to locate a point in space. But I also presented examples in Chapters 15 and 16 showing that ten-dimensional theories sometimes have the same physical consequences as eleven-dimensional theories.

Such duality suggests that our notion of dimension isn’t quite as firm as it looks—there’s a plasticity in the definition that eludes the conventional terminology. Dual descriptions of a single theory tell us that no single formulation is necessarily the best one. The formulation and even the number of dimensions in the best description might depend on the strength of the string coupling, for example. Because no single theory is always the best description, the question of the number of dimensions doesn’t always have a simple answer. This ambiguity in the meaning of dimensions and the apparent emergence of an additional dimension in strongly interacting theories are among the most important theoretical physics observations of the last decade. Let me now list a few more intriguing recent theoretical discoveries that indicate that the notion of dimension is somewhat fuzzier than we’d maybe like to believe.

I. Warped Geometry and Duality

In Chapters 20 and 22, I explained some of the consequences of the warped spacetime geometry that Raman Sundrum and I developed. In that geometry, the masses and sizes of objects depend on location along a fifth dimension and, furthermore, gravity is localized in the vicinity of a brane. But there is still one more amazing feature of this warped spacetime, known technically as anti de Sitter space, that I have yet to tell you about—one that leads to further questions about dimensionality.

The remaining remarkable feature of anti de Sitter space is the existence of a dual four-dimensional theory. Theoretical clues tell us that everything that happens in five-dimensional anti de Sitter space can be described using a dual four-dimensional framework in which there are extremely strong forces that have special properties. According to this mysterious duality, everything in the five-dimensional theory has an analog in the four-dimensional theory. And vice versa.

Although mathematical reasoning tells us that a five-dimensional theory in anti de Sitter space is equivalent to a four-dimensional one, we don’t always know the precise particle content of that four-dimensional dual theory. However, Juan Maldacena, an Argentinian-born string theorist now at the Institute for Advanced Study in Princeton, triggered a string theory frenzy in 1997 when he derived an explicit example of a similar duality in string theory. He realized that a version of string theory with a large number of overlapping D-branes on which strings interact strongly can be described either with a four-dimensional quantum field theory or with a ten-dimensional gravitational theory in which five of the ten dimensions are rolled up and the remaining five are in anti de Sitter space.

How can a four-dimensional and a five-(or ten-)dimensional theory have the same physical implications? What is the analog of an object traveling through the fifth dimension, for example? The answer is that an object moving through the fifth dimension would appear in the dual four-dimensional theory as an object that grows or shrinks. This is just like Athena’s shadow on the Gravitybrane, which grew as
she moved away from the Gravitybrane across the fifth dimension. Furthermore, objects moving past each other along the fifth dimension correspond to objects that grow and shrink and overlap in four dimensions.

Once you introduce branes, the consequences of the duality are even stranger. For example, five-dimensional anti de Sitter space with gravity but without branes is equivalent to a four-dimensional theory without gravity. But once you include a brane in the five-dimensional theory, as Raman and I did, the equivalent four-dimensional theory suddenly contains gravity.

Does this duality mean that I was cheating when I said that the warped geometries were higher-dimensional theories? Absolutely not. The duality is intriguing, but it doesn’t really change anything I’ve told you. Even if someone finds the precise dual four-dimensional theory, such a theory will be extremely difficult to study. It has to contain an enormous number of particles and such extremely strong interactions that perturbation theory (see Chapter 15) wouldn’t apply.

Theories in which objects strongly interact are almost always impossible to interpret without an alternate, weakly interacting description. And in this case, that tractable description is the five-dimensional theory. Only the five-dimensional theory has a simple enough formulation to use for computation, so it makes sense to think of the theory in five-dimensional terms. Nonetheless, even if the five-dimensional theory is more tractable, duality still makes me wonder what the word “dimensions” really means. We know that the number of dimensions should be the number of quantities you need to specify the location of an object. But are we always sure we know which quantities to count?

II. T-duality

Another reason to question the meaning of dimensions is an equivalence between two superficially different geometries that is known as
T-duality
. Even before string theorists discovered any of the dualities I’ve discussed, they discovered T-duality, which exchanges a space with a tiny rolled-up dimension for another space with a huge
rolled-up dimension.
³⁹
Odd as it may seem, in string theory, extremely small and extremely large rolled-up dimensions yield the same physical consequences. A minuscule tiny volume of rolled-up space has the same physical consequences as an extremely large one.

T-duality applies in string theory with curled-up dimensions because there are two different types of closed string in spacetime compactified on a circle, and these two strings get interchanged when a space with a tiny rolled-up dimension is exchanged for a space with a large one. The first type of closed string oscillates up and down as it circles the closed dimension, similar to the behavior of the Kaluza-Klein particles we looked at in Chapter 18. The other type wraps around the curled-up dimension. It can do so once, twice, or any number of times. And T-duality operations, which interchange a small rolled-up space for a large one, exchange these two types of string.

In fact, T-duality was the first clue that branes had to exist: without them, open strings wouldn’t have had analogs in the dual theory. But if T-duality does apply and a minuscule rolled-up dimension yields the same physical consequences as an enormous rolled-up dimension, it would mean that, once again, our notion of “dimension” is inadequate.

That is because if you imagine making the radius of one rolled-up dimension infinitely large, the T-dual rolled-up dimension would be a circle of zero size—there would be no circle at all. That is, an infinite dimension in one theory is T-dual to a theory with one dimension fewer (since a zero-size circle doesn’t count as a dimension). So T-duality also shows that two apparently different spaces could appear to have a different number of large extended dimensions, yet make identical physical predictions. Once again, the meaning of dimension is ambiguous.

III. Mirror Symmetry

T-duality applies when a dimension is rolled up into a circle. But an even weirder symmetry than T-duality is
mirror symmetry
, which sometimes applies in string theory when six dimensions are rolled up into a Calabi-Yau manifold. Mirror symmetry says that
six dimensions can be curled up into two very different Calabi-Yau manifolds, yet the resulting four-dimensional long-distance theory can be the same. The mirror manifold of a given Calabi-Yau manifold could look entirely different: it might have different shape, size, twisting, or even number of holes.
^*
Yet when there exists a mirror to a given Calabi-Yau manifold, the physical theory where six of the dimensions are curled up into either one of the two will be the same. So with mirror manifolds as well, two apparently different geometries give rise to the same predictions. Once again, spacetime has mysterious properties.

IV. Matrix Theory

Matrix theory
, a tool for studying string theory, provides still more mysterious clues about dimensions. Superficially, matrix theory looks like a quantum mechanical theory that describes the behavior and interactions of Do-branes (pointlike branes) moving through ten dimensions. But even though the theory doesn’t explicitly contain gravity, the Do-branes act like gravitons. So the theory ends up having gravitational interactions, even though the graviton is superficially absent.

Furthermore, the theory of Do-branes mimics supergravity in eleven dimensions, not ten. That is, the matrix model looks as if it contains supergravity with one more dimension than the original theory seems to describe. This suggestive behavior (along with other mathematical evidence) has led string theorists to believe that matrix theory is equivalent to M-theory, which also contains eleven-dimensional supergravity.

One especially bizarre feature of matrix theory is Edward Witten’s observation that when Do-branes come too close to each other, you can no longer know exactly where they are. As Tom Banks, Willy Fischler, Steve Shenker, and Lenny Susskind—the originators of matrix theory—said in their paper, “Thus for small distances there is
no representation of the configuration space in terms of ordinary positions.”
^*
That is, the location of a Do-brane is no longer a meaningful mathematical quantity when you try to define it too precisely.

Although such strange properties make matrix theory very tantalizing to study, it is presently very difficult to use it for computations. The problem is that—like nearly all other theories containing strongly interacting objects—no one has yet found a way to solve many of the most important questions that will help us better to understand what is really going on. Even so, because of the emergence of an extra dimension and the disappearance of dimensions when Do-branes come too close together, matrix theory is one more reason to wonder what dimensions really mean.

What to Think?

Although physicists have mathematically demonstrated these mysterious equivalences between theories with different numbers of dimensions, we are clearly still missing the big picture. Do we know with certainty that these dualities apply, and if so, what they tell us about the nature of space and time? Moreover, no one knows what the best description would be when a dimension is neither very big nor very small (relative to the extraordinarily tiny Planck scale length). Perhaps our notion of spacetime breaks down altogether once we try to describe something so small.

One of the strongest reasons for believing that our spacetime description is inadequate at the Planck scale length is that we don’t know any way, even in theory, to examine such a short distance. We know from quantum mechanics that it takes a lot of energy to investigate small length scales. But once you put too much energy into a region as small as the Planck scale length, 10
^-33
cm, you get a black hole. You then have no way to know what’s happening inside. All that information is trapped within the black hole’s event horizon.

On top of that, even if you were to try to cram more energy into
that tiny region, you wouldn’t succeed. Once you’ve put that much energy inside the Planck scale length, you can’t add any more without the region expanding. That is, the black hole would grow if you added energy. So rather than making a nice tiny probe to study that distance, you would blow the region up into something bigger and never get to study it while it’s small. It would be like trying to study delicate artifacts in a museum with a fine laser beam that instead burns them up. Even in physics thought experiments, you simply never see a region that is very much tinier than the Planck scale length. The rules of physics that we know break down before you get there. Somewhere in the vicinity of the Planck scale, conventional notions of spacetime almost certainly do not apply.