The Fabric of the Cosmos: Space, Time, and the Texture of Reality

by Brian Greene

Klein’s contribution was to suggest that what’s true for an object within the universe might be true for the fabric of the universe itself. Namely, just as the tightrope’s surface has both large and small dimensions, so does the fabric of space.

Because of its tiny size, Klein argued, the dimension would be hidden.

String Theory and Hidden Dimensions

is not as though Kaluza was led to the idea of a new spatial dimension by a rigid chain of deductive reasoning. Instead, he pulled the idea out of a hat, and upon analyzing its implications discovered an unexpected link between general relativity and electromagnetism.

The Shape of Hidden Dimensions

In the figures above, we focused on the simplest of shapes—circles, hollow spheres, solid balls—but the equations of string theory pick out a significantly more complicated class of six-dimensional shapes known as Calabi-Yau shapes or Calabi-Yau spaces. These shapes are named after two mathematicians, Eugenio Calabi and Shing-Tung Yau, who discovered them mathematically long before their relevance to string theory was realized; a rough illustration of one example is given in Figure 12.9a.

String Physics and Extra Dimensions

Why does string theory require ten spacetime dimensions?

with nine space dimensions, the constraint on the number of vibrational patterns is satisfied perfectly. And that’s how string theory determines the number of space dimensions.

since a string’s vibrational pattern determines its mass and charge, this means that the extra dimensions play a pivotal role in determining particle properties.

Both the number of extra dimensions and the geometry of each determines the properties of a particle.

The precise size and shape of the extra dimensions has a profound impact on string vibrational patterns and hence on particle properties.

The question, then, is which Calabi-Yau shape, if any, constitutes the extra-dimensional part of the spacetime fabric.

To date, the question remains unanswered. The reason is that the current understanding of string theory’s equations provides no insight into how to pick one shape from the many; from the point of view of the known equations, each Calabi-Yau shape is as valid as any other. The equations don’t even determine the size of the extra dimensions. Since we don’t see the extra dimensions, they must be small, but precisely how small remains an open question.

If a huge collection of strings all vibrate in just the right coordinated way throughout all of space, they can provide a uniform background that for all intents and purposes would be indistinguishable from a Higgs ocean. String vibrations that initially yielded zero mass would then acquire tiny nonzero masses through the drag force they experience as they move and vibrate through the string theory version of the Higgs ocean.

In the string theory version, the drag force—and hence the masses of the vibrational patterns—would be traced back to interactions between strings (since the Higgs ocean would be made of strings) and should be calculable. String theory, at least in principle, allows all particle properties to be determined by the theory itself.

The Fabric of the Cosmos According to String Theory

The Universe on a Brane

SPECULATIONS ON SPACE AND TIME IN M-THEORY

Special relativity has the constancy of the speed of light. General relativity has the equivalence principle. Quantum mechanics has the uncertainty principle. String theorists continue to grope for an analogous principle that would capture the theory’s essence as completely.

The goal of string theory—the unification of all forces and all matter in a quantum mechanical framework—is about as grand as it gets, but the theory’s evolution has been distinctly fragmented.

The Second Superstring Revolution

Over the last three decades, not one but five distinct versions of string theory have been developed. While their names are not of the essence, they are called Type I, Type IIA, Type IIB, Heterotic-O, and Heterotic-E.

Witten showed that rather than being distinct, the five theories are actually just five different ways of mathematically analyzing a single theory.

The unifying master theory has tentatively been called M-theory, M being a tantalizing placeholder whose meaning—Master? Majestic? Mother? Magic? Mystery? Matrix?—awaits the outcome of a vigorous worldwide research effort now seeking to complete the new vision illuminated by Witten’s powerful insight.

M-theory links together and embraces equally all five string theories by showing that each is part of a grander theoretical synthesis.

The Power of Translation

We have learned much about M-theory in the last few years, but we still have far to go before anyone could sensibly claim that it is properly or completely understood. In string theory, it’s as if we have five translations of a yet-to-be-discovered master text.

Theorists have found that for certain questions, one of the five may give a transparent description of the physical implications, while the descriptions given by the other four are too mathematically complex to be useful.

Witten’s work showed that each such question admits four mathematical translations—four mathematical reformulations—and sometimes one of the reformulated questions proves far simpler to answer. Thus, the dictionary for translating between the five theories can sometimes provide a means for translating impossibly difficult questions into comparatively simple ones.

Eleven Dimensions

the universe according to M-theory has ten space dimensions, that is, eleven spacetime dimensions.

Thus, Witten showed that the five ten-dimensional frameworks that string theorists had developed for more than a decade were actually five approximate descriptions of a single, underlying eleven-dimensional theory.

The collective insights of a number of researchers—Witten, Duff, Hull, Townsend, and many others—established that string theory is not just a theory of strings.

Branes

The analyses showed that there are two-dimensional objects called, naturally enough, membranes (another possible meaning for the “M” in M-theory) or—in deference to systematically naming their higher-dimensional cousins—two-branes. There are objects with three spatial dimensions called three-branes. And, although increasingly difficult to visualize, the analyses showed that there are also objects with p spatial dimensions, where p can be any whole number less than 10, known—with no derogation intended—as p-branes. Thus, strings are but one ingredient in string theory, not the ingredient.

Braneworlds

By pumping enough energy into a string, you could even make it grow to macroscopic size. With today’s technology we couldn’t come anywhere near achieving this, but it’s possible that in the searingly hot, extremely energetic aftermath of the big bang, long strings were produced. If some have managed to survive until today, they could very well stretch clear across the sky. Although a long shot, it’s even possible that such long strings could leave tiny but detectable imprints on the data we receive from space, perhaps allowing string theory to be confirmed one day through astronomical observations.

Might we, right now, be living within a three-brane?

Sticky Branes and Vibrating Strings

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.

If the endpoints of open strings are stuck within a particular region of space, what is it that they are stuck to?

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.

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).

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.

Our Universe as a Brane

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.

What about the other three? Can they probe into the extra dimensions, thus enabling us to reveal their existence?

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 ...

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.