by Brian Greene
EXPERIMENT/OBSERVATION: Gravitational Waves
EXPLANATION: Although strings are tiny, if you could somehow grab hold of one, you could stretch it large. You’d need to apply a force in excess of 1020 tons, but stretching a string is merely a matter of exerting enough energy. Theorists have found exotic situations in which the energy for such stretching might be provided by astrophysical processes, generating long strings wafting through space. Even if they were very distant, these strings might be detectable. Calculations show that as a long string vibrates, it creates ripples in spacetime—known as gravitational waves—of a highly distinctive shape, and hence they offer a clear observational signature. Within the next few decades, if not sooner, highly sensitive detectors based on earth and, funding permitting, in space, may be able to measure these ripples.
EXPERIMENT/OBSERVATION: Cosmic Microwave Background Radiation
EXPLANATION: The cosmic microwave background radiation has already proved itself capable of probing quantum physics: the measured temperature differences in the radiation arise from quantum jitters stretched large by spatial expansion. (Recall the analogy of a tiny message scribbled on a shriveled balloon becoming visible once the balloon is inflated.) In inflation, the stretching of space is so enormous that even tinier imprints, perhaps laid down by strings, might also be stretched sufficiently to be detectable—perhaps by the European Space Agency’s Planck satellite. Success or failure turns on details of how strings would have behaved in the earliest moments of the universe—the nature of the message they would have imprinted on the deflated cosmic balloon. Various ideas have been developed and calculations made. Theorists are now waiting for the data to speak for themselves.
Negative experimental results would provide much less useful information. The failure to find supersymmetric particles might mean they don’t exist, but it also might mean they are too heavy for even the Large Hadron Collider to produce; the failure to find evidence for extra dimensions might mean they don’t exist, but it also might mean they are too small for our technologies to access; the failure to find microscopic black holes might mean that gravity does not get stronger on short scales, but it also might mean that our accelerators are too weak to burrow deeply enough into the microscopic terrain where the increase in strength is substantial; the failure to find stringy signatures in observations of gravitational waves or the cosmic microwave background radiation might mean string theory is wrong, but it might also mean that the signatures are too meager for current equipment to measure.
As of today, then, the most promising positive experimental results would most likely not be able to definitively prove string theory right, while negative results would most likely not be able to prove string theory wrong.14 Yet, make no mistake. If we find evidence of extra dimensions, supersymmetry, mini black holes, or any of the other potential signatures, that will be a huge moment in the search for a unified theory. It would bolster confidence, and justifiably so, that the mathematical road we’ve been paving is headed in the right direction.
String Theory, Singularities, and Black Holes
In the vast majority of situations, quantum mechanics and gravity happily ignore each other, the former applying to small things like molecules and atoms and the latter to big things like stars and galaxies. But the two theories are forced to shed their isolation in the realms known as singularities. A singularity is any physical setting, real or hypothetical, that is so extreme (huge mass, small size, enormous spacetime curvature, punctures or rips in the spacetime fabric) that quantum mechanics and general relativity go haywire, generating results akin to the error message displayed on a calculator when you divide any number by zero.
A prize achievement of any purported quantum theory of gravity is to meld quantum mechanics and gravity in a manner that cures singularities. The resulting mathematics should never break down—even at the moment of the big bang or in the center of a black hole,15 thus providing a sensible description of situations that have long baffled researchers. It is here that string theory has made its most impressive strides, taming a growing list of singularities.
In the mid-1980s, the team of Lance Dixon, Jeff Harvey, Cumrun Vafa, and Edward Witten realized that certain punctures in the spatial fabric (known as orbifold singularities), which leave Einstein’s mathematics in shambles, pose no problem for string theory. The key to this success is that whereas point particles can fall into punctures, strings can’t. Because strings are extended objects, they can bang into a puncture, they can wrap around it, or they can get stuck to it, but these mild interactions leave the equations of string theory perfectly sound. This is important not because such ruptures in space actually happen—they may or may not—but because string theory is delivering just what we want from a quantum theory of gravity: a means of making sense of a situation that lies beyond what general relativity and quantum mechanics can handle on their own.
In the 1990s, work I did with Paul Aspinwall and David Morrison, and independent results of Edward Witten, established that yet more intense singularities (known as flop singularities) in which a spherical portion of space is compressed to an infinitesimal size can also be handled by string theory. The intuitive reasoning here is that as a string moves it can sweep across the compressed chunk of space, like a hula hoop across a soap bubble, and thus act as an encircling protective barrier. The calculations showed that such a “string shield” nullifies any potentially disastrous consequences, ensuring that string theory’s equations suffer no ill effect—no “1 divided by 0” type errors—even though the equations of conventional general relativity would fall apart.
In the years since, researchers have shown that a variety of other more complicated singularities (with names like conifolds, orientifolds, enhancons …) are also under full control within string theory. So there’s a growing list of situations that would have left Einstein, Bohr, Heisenberg, Wheeler, and Feynman saying, “We just don’t know what’s going on,” and yet for which string theory gives a complete and consistent description.
This is great progress. But a remaining challenge for string theory is to cure the singularities of black holes and the big bang, which are more severe than those so far addressed. Theorists have expended much effort trying to reach this goal, and they’ve taken significant strides. But the executive summary is that there is still a way to go before these most puzzling and most relevant of singularities are fully understood.
Nevertheless, one major advance has illuminated a related aspect of black holes. As I will discuss in Chapter 9, the work of Jacob Bekenstein and Stephen Hawking in the 1970s established that black holes contain a very particular quantity of disorder, technically known as entropy. According to basic physics, much as the disorder within a sock drawer reflects the many possible haphazard rearrangements of its contents, the disorder of a black hole reflects the many possible haphazard rearrangements of the black hole’s innards. But try as they might, physicists were unable to understand black holes well enough to identify their innards, let alone analyze the possible ways they could be rearranged. The string theorists Andrew Strominger and Cumrun Vafa broke through the impasse. Using a mélange of string theory’s fundamental ingredients (some of which we will encounter in Chapter 5), they created a mathematical model for a black hole’s disorder, a model transparent enough to enable them to extract a numerical measure of the entropy. The result they found agreed spot-on with the Bekenstein-Hawking answer. While the work left open many deep issues (such as explicitly identifying a black hole’s microscopic constituents), it provided the first firm quantum mechanical accounting of a black hole’s disorder.16
The remarkable advances in dealing with both singularities and black hole entropy give the community of physicists well-grounded confidence that in time the remaining challenges of black holes and the big bang will be conquered.
String Theory and Mathematics
Making contact with data, experimental or observational, is the only way to determine if string th
eory correctly describes nature. It’s a goal that’s proved elusive. String theory, for all its advances, is still a wholly mathematical undertaking. But string theory isn’t just a consumer of math. Some of its most important contributions have been to mathematics.
When he was developing the general theory of relativity in the early twentieth century, Einstein famously mined the mathematical archives in search of rigorous language for describing curved spacetime. The earlier geometrical insights of mathematicians such as Carl Friedrich Gauss, Bernhard Riemann, and Nikolai Lobachevsky provided an important foundation for his success. In a sense, string theory is now helping to repay Einstein’s intellectual debt by driving the development of new mathematics. There are numerous examples, but let me give one that captures the flavor of string theory’s mathematical achievements.
General relativity established a tight link between the geometry of spacetime and the physics we observe. Einstein’s equations, together with the distribution of matter and energy in a region, tell you the resulting shape of spacetime. Different physical environments (different configurations of mass and energy) yield differently shaped spacetimes; different spacetimes correspond to physically distinct environments. What would it feel like to fall into a black hole? Calculate with the spacetime geometry that Karl Schwarzschild discovered in his study of spherical solutions to Einstein’s equations. And if the black hole is rapidly spinning? Calculate with the spacetime geometry found in 1963 by the New Zealand mathematician Roy Kerr. In general relativity, geometry is the yin to physics’ yang.
String theory provides a twist to this conclusion by establishing that there can be different shapes for spacetime that nevertheless yield physically indistinguishable descriptions of reality.
Here’s one way to think about it. From antiquity to the modern mathematical era, we’ve modeled geometrical spaces as collections of points. A Ping-Pong ball, for example, is the collection of points that constitute its surface. Prior to string theory, the basic constituents making up matter were also modeled as points, point particles, and this commonality of basic ingredients spoke to an alignment between geometry and physics. But in string theory, the basic ingredient is not a point. It’s a string. This suggests that a new kind of geometry, based not on points but rather on loops, should be linked to string physics. The new geometry is called stringy geometry.
To get a feel for stringy geometry, picture a string moving through a geometrical space. Notice that the string can behave much like a point particle, innocently gliding from here to there, bumping into walls, navigating chutes and valleys, and so on. But in certain situations, a string can also do something novel. Imagine that space (or a piece of space) is shaped like a cylinder. A string can wrap itself around such a piece of space, much like a rubber band stretched around a can of soda, realizing a configuration that’s simply unavailable to a point particle. Such “wrapped” strings, and their “unwrapped” cousins, probe a geometrical space in different ways. Should a cylinder grow fatter, a string encircling it will respond by stretching, while an unwrapped string sliding on its surface won’t. In this way, wrapped and unwrapped strings are sensitive to different features of a shape through which they’re moving.
This observation is of great interest because it gives rise to a striking and thoroughly unexpected conclusion. String theorists have found special pairs of geometrical shapes for space that have completely different features when each is probed by unwrapped strings. They also have completely different features when each is probed by wrapped strings. But—and this is the punch line—when probed both ways, with wrapped and unwrapped strings, the shapes become indistinguishable. What the unwrapped strings see on one space, the wrapped strings see on the other, and vice versa, rendering identical the collective picture gleaned from the full physics of string theory.
Shapes that form such pairs provide a powerful mathematical tool. In general relativity, if you’re interested in one or another physical feature, you must complete a mathematical calculation using the unique geometrical space relevant to the situation being studied. But in string theory, the existence of pairs of physically equivalent geometrical shapes means that you have a newfound choice: you can choose to perform the necessary calculation using either shape. And the extraordinary thing is that while you’re guaranteed to get the same answer using either shape, the mathematical details en route to the answer can be vastly different. In a variety of situations, overwhelmingly difficult mathematical calculations on one geometrical shape translate into exceedingly easy calculations on the other. And any framework that makes hard mathematical calculations easy is, clearly, of great value.
Over the years, mathematicians and physicists have leveraged this hard-to-easy dictionary to make headway on a number of outstanding mathematical problems. One that I’m particularly fond of has to do with counting the number of spheres that can be packed (in a particular mathematical way) within a given Calabi-Yau shape. Mathematicians had been interested in this question for a long time but found the calculations in all but the simplest cases impenetrable. Take the Calabi-Yau shape of Figure 4.6. When a sphere is packed into this shape, it can wrap around a portion of the Calabi-Yau multiple times, much like a lasso can wrap multiple times around a beer barrel. So, how many ways can you pack a sphere into this shape if it wraps around, say, five times? When asked a question like this, mathematicians had to clear their throats, glance at their shoes, and quickly depart for pressing appointments. String theory flattened the hurdles. By translating such calculations into far easier ones on a paired Calabi-Yau shape, string theorists produced answers that knocked mathematicians back on their heels. The number of five-times-wrapped spheres packed into the Calabi-Yau in Figure 4.6? 229,305,888,887,625. And if the spheres wrap around themselves ten times? . Twenty times? . These numbers proved to be harbingers for a spectrum of results that have opened a whole new chapter in mathematical discovery.17
So, whether or not string theory offers a correct approach to describing the physical universe, it has already established itself as a potent tool for investigating the mathematical one.
The State of String Theory: An Evaluation
Building on the last four sections, Table 4.2 provides a status report for string theory, including some additional observations that I didn’t explicitly call out in the text above. It paints a picture of a theory in progress, one that has produced stunning achievements but has not yet been tested on the most important scale: experimental confirmation. The theory will remain speculative until a convincing link to experiment or observation is forged. Establishing such a link is the great challenge. But it’s not a challenge that’s peculiar to string theory. Any attempt to unite gravity and quantum mechanics enters a domain that’s far beyond the cutting-edge of experimental research. It’s part and parcel of taking on such a supremely ambitious goal. Pushing the fundamental boundaries of knowledge, seeking answers to some of the deepest questions contemplated during the past few thousand years of human thought, is a formidable undertaking, one that won’t likely be completed overnight. Nor in a handful of decades.
In evaluating the state of the art, many string theorists argue that a crucial next step is to articulate the theory’s equations in their most exact, useful, and comprehensive form. Much of the research during the theory’s first couple of decades, through the mid-1990s, was carried out using approximate equations that many were convinced could reveal the theory’s gross features but were too coarse to yield refined predictions. Recent advances, to which we will now turn, have catapulted understanding far beyond what could be achieved by the approximate methods. While definitive predictions have remained elusive, a new perspective has emerged. It’s come from a series of breakthroughs that has opened grand new vistas on the theory’s potential implications, among which are new varieties of parallel worlds.
Table 4.2. A summary status report for string theory.
GOAL: Unite gravity and quantum mechanics
IS GOAL REQUIR
ED?: Yes.
The primary goal is to meld general relativity and quantum mechanics.
STATUS: Excellent.
A wealth of calculations and insights attest to string theory’s successful merger of general relativity and quantum mechanics.18
GOAL: Unify all forces
IS GOAL REQUIRED?: No.
Unification of gravity and quantum mechanics does not require a further unification with the other forces of nature.
STATUS: Excellent.
While not required, a fully unified theory has long been a goal of physics research. String theory achieves this goal by describing all forces in the same manner—their quanta are strings executing particular vibrational patterns.
GOAL: Incorporate key breakthroughs from past research
IS GOAL REQUIRED?: No.
In principle, a successful theory need bear little resemblance to successful theories from the past.
STATUS: Excellent.
Though progress isn’t necessarily incremental, history shows that it usually is; successful new theories typically embrace past successes as limiting cases. String theory incorporates the essential key breakthroughs from previously successful physical frameworks.
GOAL: Explain particle properties
IS GOAL REQUIRED?: No.
This is a noble goal, and if achieved would provide a profound level of explanation—but it is not required of a successful theory of quantum gravity.
STATUS: Indeterminate; no predictions.
Going beyond quantum field theory, string theory offers a framework for explaining particle properties. But to date, this potential remains unrealized; the many different possible forms for the extra dimensions imply many different possible collections of particle properties. There is no currently available means to pick one shape from the many.