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THE FABRIC of the COSMOS, Brian Greene, 2004
```(annotated and with added bold highlights by Epsilon=One)
```(annotated and with added bold highlights by Epsilon=One)
Chapter 13 - The Universe on a Brane
Branes
A natural question, which may have occurred to you in the last chapter, is Why strings? Why are one-dimensional ingredients so special? In reconciling quantum mechanics and general relativity, we found it crucial that strings are not dots, that they have nonzero size. But that requirement can be met with two-dimensional ingredients shaped like miniature disks or Frisbees, or by three-dimensional bloblike ingredients, shaped like baseballs or lumps of clay. Or, since the theory has such an abundance of space dimensions, we can even imagine blobs with more dimensions still. Why don't these ingredients play any role in our fundamental theories?
In the 1980s and early 1990s, most string theorists had what seemed like a convincing answer. They argued that there had been attempts to formulate a fundamental theory of matter based on bloblike constituents by, among others, such icons of twentieth-century physics as Werner Heisenberg and Paul Dirac. But their work, as well as many subsequent studies, showed that it was extremely difficult to develop a theory based on tiny blobs that met the most basic of physical requirements — for example, ensuring that all quantum mechanical probabilities lie between 0 and 1 (no sense can be made of negative probabilities or of probabilities greater than 1), and debarring faster-than-light communication. For point particles, a half-century of research initiated in the 1920s showed that these conditions could be met (as long as gravity was ignored). And, by the 1980s, more than a decade of investigation by Schwarz, Scherk, Green, and others established, to the surprise of most researchers, that the conditions could also be met for one-dimensional ingredients, strings (which necessarily included gravity). But it seemed impossible to proceed to fundamental ingredients with two or more spatial dimensions. The reason briefly put, is that the number of symmetries respected by the equations peaks enormously for one-dimensional objects (strings) and drops off precipitously thereafter. The symmetries in question are more abstract than the ones discussed in Chapter 8 (they have to do with how equations change if, while studying the motion of a string or a higher dimensional ingredient, we were to zoom in or out, suddenly and arbitrarily changing the resolution of our observations). These transformations prove critical to formulating a physically sensible set of equations, and beyond strings it seemed that the required fecundity of symmetries was absent. 1
It was thus another shock to most string theorists when Witten's paper and an avalanche of subsequent results 2 led to the realization that string theory, and the M-theoretic framework to which it now belongs, does contain ingredients besides strings. 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 also 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.
These other ingredients escaped earlier theoretical investigation for much the same reason the tenth space dimension did: the approximate string equations proved too coarse to reveal them. In the theoretical contexts that string researchers had investigated mathematically, it turns out that all p-branes are significantly heavier than strings. And the more massive something is, the more energy is required to produce it. But a limitation of the approximate string equations — a limitation embedded in the equations and well known to all string theorists — is that they become less and less accurate when describing entities and processes involving more and more energy. At the extreme energies relevant for p-branes, the approximate equations lacked the precision to expose the branes lurking in the shadows, and that's why decades passed without their presence being noticed in the mathematics. But with the various rephrasings and new approaches provided by the unified M-theoretic framework, researchers were able to skirt some of the previous technical obstacles, and there, in full mathematical view, they found a whole panoply of higher-dimensional ingredients. 3
The revelation that there are other ingredients besides strings in string theory does not invalidate or make obsolete earlier work any more than the discovery of the tenth spatial dimension did. Research shows that if the higher-dimensional branes are much more massive than strings — as had been unknowingly assumed in previous studies — they have minimal impact on a wide range of theoretical calculations. But just as the tenth space dimension does not have to be much smaller than all others, so the higher-dimensional branes do not have to be much heavier. There are a variety of circumstances, still hypothetical, in which the mass of a higher-dimensional brane can be on a par with the lowest-mass string vibrational patterns, and in such cases the brane does have a significant impact on the resulting physics. For example, my own work with Andrew Strominger and David Morrison showed that a brane can wrap itself around a spherical portion of a Calabi-Yau shape, much like plastic wrap vacuum-sealed around a grapefruit; should that portion of space shrink, the wrapped brane would also shrink, causing its mass to decrease. This decrease in mass, we were able to show, allows the portion of space to collapse fully and tear open — space itself can rip apart — while the wrapped brane ensures that there are no catastrophic physical consequences. I discussed this development in detail in The Elegant Universe and will briefly return to it when we discuss time travel in Chapter 15, so I won't elaborate further here. But this snippet makes clear how higher-dimensional branes can have a significant effect on the physics of string theory.
For our current focus, though, there is another profound way that branes impact the view of the universe according to string/M-theory. The grand expanse of the cosmos — the entirety of the spacetime of which we are aware — may itself be nothing but an enormous brane. Ours may be a braneworld.
In the 1980s and early 1990s, most string theorists had what seemed like a convincing answer. They argued that there had been attempts to formulate a fundamental theory of matter based on bloblike constituents by, among others, such icons of twentieth-century physics as Werner Heisenberg and Paul Dirac. But their work, as well as many subsequent studies, showed that it was extremely difficult to develop a theory based on tiny blobs that met the most basic of physical requirements — for example, ensuring that all quantum mechanical probabilities lie between 0 and 1 (no sense can be made of negative probabilities or of probabilities greater than 1), and debarring faster-than-light communication. For point particles, a half-century of research initiated in the 1920s showed that these conditions could be met (as long as gravity was ignored). And, by the 1980s, more than a decade of investigation by Schwarz, Scherk, Green, and others established, to the surprise of most researchers, that the conditions could also be met for one-dimensional ingredients, strings (which necessarily included gravity). But it seemed impossible to proceed to fundamental ingredients with two or more spatial dimensions. The reason briefly put, is that the number of symmetries respected by the equations peaks enormously for one-dimensional objects (strings) and drops off precipitously thereafter. The symmetries in question are more abstract than the ones discussed in Chapter 8 (they have to do with how equations change if, while studying the motion of a string or a higher dimensional ingredient, we were to zoom in or out, suddenly and arbitrarily changing the resolution of our observations). These transformations prove critical to formulating a physically sensible set of equations, and beyond strings it seemed that the required fecundity of symmetries was absent. 1
It was thus another shock to most string theorists when Witten's paper and an avalanche of subsequent results 2 led to the realization that string theory, and the M-theoretic framework to which it now belongs, does contain ingredients besides strings. 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 also 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.
These other ingredients escaped earlier theoretical investigation for much the same reason the tenth space dimension did: the approximate string equations proved too coarse to reveal them. In the theoretical contexts that string researchers had investigated mathematically, it turns out that all p-branes are significantly heavier than strings. And the more massive something is, the more energy is required to produce it. But a limitation of the approximate string equations — a limitation embedded in the equations and well known to all string theorists — is that they become less and less accurate when describing entities and processes involving more and more energy. At the extreme energies relevant for p-branes, the approximate equations lacked the precision to expose the branes lurking in the shadows, and that's why decades passed without their presence being noticed in the mathematics. But with the various rephrasings and new approaches provided by the unified M-theoretic framework, researchers were able to skirt some of the previous technical obstacles, and there, in full mathematical view, they found a whole panoply of higher-dimensional ingredients. 3
The revelation that there are other ingredients besides strings in string theory does not invalidate or make obsolete earlier work any more than the discovery of the tenth spatial dimension did. Research shows that if the higher-dimensional branes are much more massive than strings — as had been unknowingly assumed in previous studies — they have minimal impact on a wide range of theoretical calculations. But just as the tenth space dimension does not have to be much smaller than all others, so the higher-dimensional branes do not have to be much heavier. There are a variety of circumstances, still hypothetical, in which the mass of a higher-dimensional brane can be on a par with the lowest-mass string vibrational patterns, and in such cases the brane does have a significant impact on the resulting physics. For example, my own work with Andrew Strominger and David Morrison showed that a brane can wrap itself around a spherical portion of a Calabi-Yau shape, much like plastic wrap vacuum-sealed around a grapefruit; should that portion of space shrink, the wrapped brane would also shrink, causing its mass to decrease. This decrease in mass, we were able to show, allows the portion of space to collapse fully and tear open — space itself can rip apart — while the wrapped brane ensures that there are no catastrophic physical consequences. I discussed this development in detail in The Elegant Universe and will briefly return to it when we discuss time travel in Chapter 15, so I won't elaborate further here. But this snippet makes clear how higher-dimensional branes can have a significant effect on the physics of string theory.
For our current focus, though, there is another profound way that branes impact the view of the universe according to string/M-theory. The grand expanse of the cosmos — the entirety of the spacetime of which we are aware — may itself be nothing but an enormous brane. Ours may be a braneworld.