The Fabric of the Cosmos: Space, Time, and the Texture of Reality (50 page)

One of the motivations for introducing the term "M-theory" is that we now realize that "string theory" highlights but one of the theory's many ingredients. Theoretical studies revealed one-dimensional strings decades before more refined analyses discovered the higher-dimensional branes, so "string theory" is something of an historical artifact. But, even though M-theory exhibits a democracy in which extended objects of a variety of dimensions are represented, strings still play a central role in our current formulation. In one way this is immediately clear. When all the higher-dimensional
p-
branes are much heavier than strings, they can be ignored, as researchers had done unknowingly since the 1970s. But there is another, more general way in which strings are first among equals.

In 1995, shortly after Witten announced his breakthrough, Joe Polchinski of the University of California at Santa Barbara got to thinking. Years earlier, in a paper he had written with Robert Leigh and Jin Dai, Polchinski had discovered an interesting though fairly obscure feature of string theory. Polchinski's motivation and reasoning were somewhat technical and the details are not essential to our discussion, but his results are. He found that in certain situations the endpoints of open strings— remember, these are string segments with two loose ends—would not be able to move with complete freedom. Instead, just as a bead on a wire is free to move, but must follow the wire's contour, and just as a pinball is free to move, but must follow the contours of the pinball table's surface, the endpoints of an open string would be free to move but would be restricted to particular shapes or contours in space. While the string would still be free to vibrate, Polchinski and his collaborators showed that its endpoints would be "stuck" or "trapped" within certain regions.

In some situations, the region might be one-dimensional, in which case the string's endpoints would be like two beads sliding on a wire, with the string itself being like a cord connecting them. In other situations, the region might be two-dimensional, in which case the endpoints of the string would be very much like two pinballs connected by a cord, rolling around a pinball table. In yet other situations, the region might have three, four, or any other number of spatial dimensions less than ten. These results, as shown by Polchinski and also by Petr Ho ava and Michael Green, helped resolve a long-standing puzzle in the comparison of open and closed strings, but over the years, the work attracted limited attention.
⁵
In October 1995, when Polchinski finished rethinking these earlier insights in light of Witten's new discoveries, that changed.

A question that Polchinski's earlier paper left without a complete answer is one that may have occurred to you while reading the last paragraph: If the endpoints of open strings are stuck within a particular region of space,
what is it that they are stuck to
? Wires and pinball machines have a tangible existence independent of the beads or balls that are constrained to move along them. What about the regions of space to which the endpoints of open strings are constrained? Are they filled with some independent and fundamental ingredient of string theory, one that jealously clutches open string endpoints? Prior to 1995, when string theory was thought to be a theory of strings only, there didn't seem to be any candidate for the job. But after Witten's breakthrough and the torrent of results it inspired, the answer became obvious to Polchinski: if the endpoints of open strings are restricted to move within some
p-
dimensional region of space, then that region of space must be occupied by a
p-
brane.
³⁷
His calculations showed that the newly discovered
p-
branes had exactly the right properties to be the objects that exert an unbreakable grip on open string endpoints, constraining them to move within the
p-
dimensional region of space they fill.

To get a better sense for what this means, look at Figure 13.2. In (a), we see a couple of two-branes with a slew of open strings moving around and vibrating, all with their endpoints restricted to motion along their respective branes. Although it is increasingly difficult to draw, the situation with higher-dimensional branes is identical. Open string endpoints can move freely on and within the
p-
brane, but they can't leave the brane itself. When it comes to the possibility of motion off a brane, branes are the stickiest things imaginable. It's also possible for one end of an open string to be stuck to one
p-
brane and its other end to be stuck to a different
p-
brane, one that may have the same dimension as the first (Figure 13.2b) or may not (Figure 13.2c).

To Witten's discovery of the connection between the various string theories, Polchinski's new paper provided a companion manifesto for the second superstring revolution. While some of the great minds of twentieth-century theoretical physics had struggled and failed to formulate a theory containing fundamental ingredients with more dimensions than dots (zero dimensions) or strings (one dimension), the results of Witten and Polchinski, together with important insights of many of today's leading researchers, revealed the path to progress. Not only did these physicists establish that string/M-theory contains higher-dimensional ingredients, but Polchinski's insights in particular provided a means for analyzing their detailed physical properties theoretically (should they prove to exist). The properties of a brane, Polchinski argued, are to a large extent captured by the properties of the vibrating open strings whose endpoints it contains. Just as you can learn a lot about a carpet by running your hand through its pile—the snippets of wool whose endpoints are attached to the carpet backing—many qualities of a brane can be determined by studying the strings whose endpoints it clutches.

Figure 13.2
(
a
)
Open strings with endpoints attached to two-dimensional branes, or two-branes.
(
b
)
Strings stretching from one two-brane to another.
(
c
)
Strings stretching from a two-brane to a one-brane.

That was a paramount result. It showed that decades of research that produced sharp mathematical methods to study one-dimensional objects—strings—could be used to study higher-dimensional objects,
p-
branes. Wonderfully, then, Polchinski revealed that the analysis of higher-dimensional objects was reduced, to a large degree, to the thoroughly familiar, if still hypothetical, analysis of strings. It's in this sense that strings are special among equals. If you understand the behavior of strings, you're a long way toward understanding the behavior of
p-
branes.

With these insights, let's now return to the braneworld scenario—the possibility that we're all living out our lives within a three-brane.

If we are living within a three-brane—if our four-dimensional spacetime is nothing but the history swept out by a three-brane through time—then the venerable question of whether spacetime is a something would be cast in a brilliant new light. Familiar four-dimensional spacetime would arise from a real physical entity in string/M-theory, a three-brane, not from some vague or abstract idea. In this approach, the reality of our four-dimensional spacetime would be on a par with the reality of an electron or a quark. (Of course, you could still ask whether the larger spacetime within which strings and branes exist—the eleven dimensions of string/M-theory—is itself an entity; the reality of the spacetime arena we directly experience, though, would be rendered obvious.) But if the universe we're aware of really is a three-brane, wouldn't even a casual glance reveal that we are immersed within something—within the three-brane interior?

Well, we've already learned of things within which modern physics suggests we may be immersed—a Higgs ocean; space filled with dark energy; myriad quantum field fluctuations—none of which make themselves directly apparent to unaided human perceptions. So it shouldn't be a shock to learn that string/M-theory adds another candidate to the list of invisible things that may fill "empty" space. But let's not get cavalier. For each of the previous possibilities, we understand its impact on physics and how we might establish that it truly exists. Indeed, for two of the three— dark energy and quantum fluctuations—we've seen that strong evidence supporting their existence has already been gathered; evidence for the Higgs field is being sought at current and future accelerators. So what is the corresponding situation for life within a three-brane? If the braneworld scenario is correct, why don't we see the three-brane, and how would we establish that it exists?

The answer highlights how the physical implications of string/M-THEORY in the braneworld context differ radically from the earlier "brane-free" (or, as they're sometimes affectionately called, no-braner) scenarios. Consider, as an important example, the motion of light—the motion of photons. In string theory, a photon, as you now know, is a particular string vibrational pattern. More specifically, mathematical studies have shown that in the braneworld scenario, only open string vibrations, not closed ones, produce photons, and this makes a big difference. Open string endpoints are constrained to move within the three-brane, but are otherwise completely free. This implies that photons (open strings executing the photon mode of vibration) would travel without any constraint or obstruction throughout our three-brane. And that would make the brane appear
completely transparent—completely invisible—
thus preventing us from seeing that we are immersed within it.

Of equal importance, because open string endpoints cannot leave a brane, they are unable to move into the extra dimensions. Just as the wire constrains its beads and the pinball machine constrains its balls, our sticky three-brane would permit photons to move
only
within our three spatial dimensions. Since photons are the messenger particles for electromagnetism, this implies that the electromagnetic force—light—would be trapped within our three dimensions, as illustrated (in two dimensions so we can draw it) in Figure 13.3.

That's an intense realization with important consequences. Earlier, we required the extra dimensions of string/M-theory to be tightly curled up. The reason, clearly, is that we don't see the extra dimensions and so they must be hidden away. And one way to hide them is to make them smaller than we or our equipment can detect. But let's now reexamine this issue in the braneworld scenario. How do we detect things? Well, when we use our eyes, we use the electromagnetic force; when we use powerful instruments like electron microscopes, we also use the electromagnetic force; when we use atom smashers, one of the forces we use to probe the ultrasmall is, once again, the electromagnetic force. But if the electromagnetic force is confined to our three-brane, our three space dimensions, it is
unable
to probe the extra dimensions, regardless of their size. Photons cannot escape our dimensions, enter the extra dimensions, and then travel back to our eyes or equipment allowing us to detect the extra dimensions,
even if they were as large as the familiar space dimensions.

Figure 13.3
(
a
)
In the braneworld scenario, photons are open strings with endpoints trapped within the brane, so they—light—cannot leave the brane itself.
(
b
)
Our braneworld could be floating in a grand expanse of additional dimensions that remain invisible to us, because the light we see cannot leave our brane. There might also be other braneworlds floating nearby.

So, if we live in a three-brane, there is an alternative explanation for why we're not aware of the extra dimensions. It is not necessarily that the extra dimensions are extremely small. They could be big. We don't see them because of the
way
we see. We see by using the electromagnetic force, which is unable to access any dimensions beyond the three we know about. Like an ant walking along a lily pad, completely unaware of the deep waters lying just beneath the visible surface, we could be floating within a grand, expansive, higher-dimensional space, as in Figure 13.3b, but the electromagnetic force—eternally trapped within our dimensions—would be unable to reveal this.

Okay, you might say, but the electromagnetic force is only one of nature's four forces. What about the other three? Can they probe into the extra dimensions, thus enabling us to reveal their existence? For the strong and weak nuclear forces, the answer is, again, no. In the braneworld scenario, calculations show that the messenger particles for these forces—gluons and W and Z particles—also arise from open-string vibrational patterns, so they are just as trapped as photons, and processes involving the strong and weak nuclear forces are just as blind to the extra dimensions. The same goes for particles of matter. Electrons, quarks, and all other particle species also arise from the vibrations of open strings with trapped endpoints.
Thus, in the braneworld scenario, you and I and
everything we've ever seen are permanently imprisoned within our three-brane.
Taking account of time, everything is trapped within our four-dimensional slice of spacetime.

Well, almost everything. For the force of gravity, the situation is different. Mathematical analyses of the braneworld scenario have shown that gravitons arise from the vibrational pattern of closed strings, much as they do in the previously discussed no-braner scenarios. And closed strings— strings with no endpoints—are not trapped by branes. They are as free to leave a brane as they are to roam on or through it. So, if we were living in a brane, we would not be completely cut off from the extra dimensions. Through the gravitational force, we could both influence and be influenced by the extra dimensions. Gravity, in such a scenario, would provide our sole means for interacting beyond our three space dimensions.

How big could the extra dimensions be before we'd become aware of them through the gravitational force? This is an interesting and critical question, so let's take a look.