George Bernhard Riemann, a nineteenth-century German mathematician, figured out how to apply geometry to curved spaces. Einstein recognized that Rienmann’s geometry accurately described the physics of gravity, and Reinmann’s theories supplied him with the necessary mathematical foundations to analyze warped space. The curvature of spacetime, Rienmann found, is expressed mathematically as the distorted distances between its points. Einstein applied Rienmann’s discovery to the physical realm and concluded that the gravitational force felt by an object directly reflects this distortion.

String theory deals in short-distance physics, and Rienmannian geometry ceases to function at an ultramicroscopic level. This means that, for string theory to work, physicists must modify both Riemannian geometry and the general theory of relativity that Einstein derived from it. A new type of geometry is necessary to decipher tiny Planck-length scales. Physicists have called this new type of geometry quantum geometry.

Fifteen billion years ago, the universe began with the big bang. As Hubble discovered, the universe is constantly expanding, which makes it hard to measure the average density of matter in the universe. If the average matter density exceeds a so-called critical density of a hundredth of a billionth of a billionth of a billionth (10–29) of a gram per cubic centimeter, then a large gravitational force will permeate the cosmos and reverse the expansion. If the average density is less than the critical density, the gravitational expansion will be too weak to do this. (The earth is not a reliable indicator for the average density of the universe: Matter clumps, and the vast empty spaces between galaxies bring the average down.)

Conventional wisdom proclaims that the universe began with a bang from an initial zero-size state. If the universe has enough mass, it will eventually end with a “crunch” that will reduce it to a similar state of compression. String theory is required to help physicists evaluate the extremely compressed early stage; it has set Planck length as the lower limit on the size of the “Big Crunch.” It would not make sense to set this same limit for the point-particle model.

To return to the garden hose analogy for the universe: strings, unlike point particles, can “lasso” the circular part of the garden hose. When a string is in this position, it is in a winding mode of motion, which is a possibility that is inherent to strings. A string in winding mode has a minimum mass that is determined by the size of the circular dimension it is wrapping around and the number of times it is wrapped.

Wound-string configurations suggest that a string’s energy comes from two sources: vibrational motion and winding energy. All string motion is a combination of sliding and oscillating. Strings’ vibrational movements have energies that are inversely proportional to the radius of the circle they are wrapping. A small radius, for example, would confine the string more strictly and would contain more energy. But the winding mode energies are directly proportional to the radius. Greene eventually explains what this means: there is no distinction between geometrically distinct forms. The same goes for total string energies: there is no distinction between different sizes for the circular dimension! Through a complicated chain of explanations, Greene shows that there is absolutely no way to differentiate between radii that are inversely related to one another.

Greene then takes the odd mirror-symmetry phenomenon and applies it to distance. Robert Brandenberger and Cumrun Vafa have shown that, when dealing with circular spatial dimensions, physicists must consider two different definitions of distance. Only one of these definitions conforms to our conventional understanding, because people generally only take into account one concept of distance. The idea that the universe is huge is one that quantum geometry calls into question. According to light string modes, the universe is huge and expanding; according to heavy string modes, it is miniscule and contracting.

This same seeming contradiction extends to the possibility of two different Calabi-Yau shapes giving rise to identical physics. Greene recounts how he and Ronen Plesser discovered mirror symmetry at almost exactly the same time that colleague Philip Candelas did. Mirror manifolds are physically indistinguishable but are geometrically distinct. Physicists originally thought that these Calabi Yau spaces were entirely unrelated, but eventually they found a way to connect them through string theory. This symmetrical pairing allows what would be a very difficult calculation of a particular Calabi-Yau space to be done on its simpler mirror-symmetrical pair. A decade after this discovery, mathematicians have made great strides in revealing the inherent mathematical foundations of mirror symmetry.

According to Einstein’s general relativity, it is impossible to tear the fabric of space. Still, many string theorists who dare to go beyond Einstein’s classical theory have wondered if the spatial fabric of the universe can indeed be ripped and torn. The discovery that quantum physics is a realm of violent turbulence has led many to think that perhaps the spatial fabric rips on a regular basis.

In this chapter, Greene introduces the concept of a wormhole as a bridge or tunnel that supplies a shortcut between different regions of the universe and in the process creates a new region of space. No one yet knows if wormholes exist, but if they do, they will provide evidence that space can indeed be stretched into fantastic contortions. Black holes are another example of space stretched to its limit, and there is a strong experimental basis for believing that black holes exist.

String theorists believe that the fabric of space can tear in specific ways. In 1987, Yau and one of his students found that a Calabi-Yau space could be changed into a different Calabi-Yau space by mathematically puncturing its surface and then “sewing up” the hole. They performed a series of mathematical manipulations called flop transitions, which means that the original Calabi-Yau space is flopped over into a new configuration. Through diagrams, Greene shows how the first Calabi-Yau space is “topologically distinct” from the second. The deformation, he says, could not have occurred without the fabric of the first Calabi-Yau space being torn at some stage.

Green then describes his work with mirror-symmetry flop transitions. Several of his colleagues tried to determine what would happen if the spatial fabric of the Calabi-Yau section of the universe underwent a flop transition. What would it look like from the perspective of the mirror Calabi-Yau space? After long trials, they concluded that mirror perspective transition could indeed take place with no catastrophic consequences.

Throughout 1992, Greene and Plesser attempted to gather mathematical evidence of mirror-perspective Calabi-Yau spaces. Greene decided to spend the fall of 1992 at the Institute for Advanced Study with mathematician David Morrison and Greene’s Oxford classmate Paul Aspinwall. Over the course of that fall, Morrison, Aspinwell, and Greene proved mathematically that flop transitions did not destroy mirror symmetry. Around the same time, Witten had also established, by different methods, that flop transitions occur in string theory. Witten went beyond Greene and his coresearchers’ findings to show why flop transitions do not trigger cosmic catastrophe: when a tear occurs, an adjacent string encircles and reconstitutes it. Together, Greene, Morrison, Aspinwall, and Witten mathematically demonstrated the existence of topology-changing transitions (a more technical name for flop transitions). These findings, Greene predicts, will lead to a revolutionary revision of Einstein’s general relativity.

This chapter is arguably the most involved in the book, and Greene recommends that readers skip some of its finer points if necessary. Greene begins by describing the many problems that have dogged string theory throughout the 1980s. Overabundance was the main concern. For most of the decade, five different versions of string theory emerged, no one more valid than any other. Also, there were too many possible Calabi-Yau shapes, too many variables, and too many approximations for any coherent answers to surface.

Greene doesn’t doubt that the exact equations will fall into place one day. Since the onset of the second superstring revolution in 1995, Witten has predicted that the five competing versions of string theory will one day be revealed as variations on the same theory, all components of the same overarching framework, which has come to be known as M-theory. More and more physicists are beginning to agree with Greene. M-theory requires eleven dimensions—ten of space and one of time. Theorists have realized that the extra spatial dimension permits the five versions of string theory to be synthesized harmoniously. Physicists had initially overlooked the eleventh dimension because their calculations were too approximate.

While M-theory contains vibrating one-dimensional strings (one-branes), it also incorporates other objects: two-dimensional membranes (two-branes), three-dimensional blobs (three-branes), and even more unexpected components. Greene believes that making sense of M-theory is the greatest challenge that physicists face in the twenty-first century.

Perturbation theory continues to set limits on physicists’ methodology. As a reminder, perturbation theory is the process by which physicists make approximations in the hopes of getting a rough answer to a question. The perturbative approach helped make sense of virtual string pairs, but no one knew if it was producing accurate answers. The string coupling constant is a positive number that determines the likelihood either that a string will split apart into two strings or that two strings will merge into one. A string coupling constant of less than one indicates weak coupling, suggesting that the perturbative method will be valid. If, however, a string coupling constant is greater than one, indicating strong coupling, perturbative theory becomes useless. Because they do not yet know the value of this constant, physicists must rely on approximations.

In 1995, Witten launched the second superstring revolution by introducing duality, a concept that authorizes the application of perturbation theory to a much wider range of problems. String theory contains many examples of duality, including the string pairs produced by mirror symmetry and the equivalence of circular-dimension string computations. Witten argued that the five different versions of string theory were all dual because each version had an equivalent string in at least one other theory.

Greene makes an unlikely comparison between black holes and elementary particles. Both, he says, have an internal structure that physicists have yet to identify. It has recently been suggested that an even greater similarity exists: perhaps black holes are actually huge elementary particles. After all, Einstein set no minimum limit on the mass of a black hole. Therefore, if we crushed a chunk of matter into ever-smaller black holes, the result would be an object no different from an elementary particle. This is because both are defined by their mass, force charges, and spin.

String theorists have long predicted the existence of three-dimensional spheres embedded in the fabric of a Calabi-Yau space, and recently they have wondered what would happen if one of those spheres were to collapse. Cosmic catastrophe? Apocalypse? Physicists previously believed that the entire universe would fall apart if such pinching of the spatial fabric occurred, but in 1995 Andrew Strominger disproved these fears. He showed that a one-brane string can completely wrap around a one-dimensional portion of space, a two-brane around a two-dimensional sphere, and a three-brane around a three-dimensional sphere. This wrapping shields the three-brane from any cataclysmic results should a three-brane collapse. Physics continues to behave even after a three-dimensional sphere shrinks into a point.

Greene elaborated on Strominger’s idea and found that when the three-dimensional sphere collapses, the Calabi-Yau space might be capable of repairing itself by reinflating the sphere. The three-dimensional sphere is replaced by a two-dimensional sphere. Greene and others showed how one Calabi-Yau space can transform into an entirely different space, with a different number of holes. This insight led them to believe that the fabric of space can be ripped and torn far more dramatically than previously imagined. These extreme space-tearing metamorphoses are called conifold transitions.

String theory predicts that black holes can undergo an analogous sort of transformation, changing into zero-mass elementary particles through what is known as a phase transition. Water offers a more easily understood example of a phase transition. Water can exist as a solid (ice), a liquid (liquid water), or a gas (steam). As improbable as it may sound, string theorists believe that black holes and photons are really just two different phases of the same stringy material.

In 1970, Jacob Bekenstein proposed the theory of black hole entropy, which is grounded in the second law of thermodymanics. Bekenstein argued that because black holes have a huge amount of entropy, their event horizon increases after every physical interaction. Most physicists doubted this claim. They believed that black holes ranked among the most orderly objects in the universe and were too simple to support disorder. Most important, entropy belonged to the conceptual framework of quantum mechanics and black holes belonged to the opposing framework of general relativity. It was impossible to discuss the entropy of a black hole without somehow merging these two unwieldy frameworks.

In 1974, Stephen Hawking attempted to confirm Bekenstein’s hypothesis by applying quantum mechanics to black holes. He successfully proved that black holes emit radiation. When photon pairs being sucked into the holes are torn apart just outside the event horizon, the blackness starts to glow. Black holes, Hawking went on to prove, do indeed have entropy and temperature. The gravitational laws they obey are extremely similar to the laws of thermodynamics. Then, in 1996, Strominger and Vafa made another huge advance when they used string theory to identify the microscopic properties of certain black holes. Their findings exactly agreed with Bekenstein and Hawking’s earlier discoveries. Strominger and Vafa even tracked how to generate a particular type of black hole from recently discovered constituents of string theory.

According to nineteenth-century French mathematician Pierre-Simon de Laplace, if you know the positions and velocities of every particle in the universe, then you can use Newton’s laws of motion to determine their position and velocity at any other time in the past or future. But Heisenberg’s uncertainty principle undermined Laplace’s classical theory of determinism. The uncertainty principle was soon supplanted by quantum determinism, which states that the probability of an event happening at a given time in the future is determined by knowledge of the wave functions at any earlier time. It was no longer possible to predict certain outcomes with any precision or confidence. In 1976, Hawking argued that the existence of black holes violated even this toned-down determinism. If an object is sucked into a black hole, then its wave function is likewise swallowed. Can any information that goes beyond a black hole’s event horizon ever re-emerge? Hawking thinks not, but string theorists are offering convincing proofs that the information might indeed resurface again. The question, like many in string theory, remains unanswered.

To sum up Greene’s basic point in this difficult chapter: only string theory locates the disorder in the high entropy of a large black hole. The existing theories, general relativity and quantum mechanics, fail to explain satisfactorily the two cosmic extremes—enormous mass and ultramicroscopic particles. Einstein’s classical theory no longer applies to objects on these scales. String theorists are currently working toward postulating a theory about the “spacetime singularity” of black holes that might resolve some of these mysteries.

Greene first outlines the pre-string-theory standard model of cosmology, which originated in the fifteen years after Einstein promulgated his general theory of relativity. The basis of this model is the big bang theory, an extremely energetic event that occurred roughly 15 billion years ago, when the universe erupted into existence. With the passage of Planck time (10–43 seconds) immediately following the bang, the universe was 10–32 degrees Kelvin, which is about 10 trillion times hotter than the deepest interior of the sun. As the universe cooled, quarks began to lump together into groups of three, forming protons and neutrons. Over the next three minutes—a period known as primordial nucleosynthesis—the majority of nuclei that emerged were hydrogen and helium. In the next few hundred thousand years, the universe continued to expand and cool. Then, when the temperature dropped enough, the first electrons slowed down enough to be trapped by the atomic nuclei. Thus, the first electrically neutral atoms emerged. Before the electrons were captured, the universe was covered with a plasma of electrically charged particles, but from this point on, it was transparent. Photons were, for the first time, able to move about uninhibited. It was approximately a billion years after the bang that galaxies, stars, and planets began to emerge.

Astronomers use powerful telescopes to verify the universe’s ever-expanding state. They discovered something strange called cosmic background radiation: microwave radiation (long-wavelength light) that has suffused the universe since just after the bang. This microwave radiation is an atmospheric relic of the meltdown that occurred. Cosmic background radiation isn’t dangerous, but the discovery of its existence—even in trace form—pointed to major gaps in scientists’ understanding of the bang. In one part of the sky, the radiation differs hardly at all from the radiation in another part of the sky. Think how strange it would be if every place on earth were exactly the same temperature all the time—Antarctica, Hawaii, Sierra Leone, anywhere. The cosmic background radiation suggests that, at some point, the universe was entirely homogenous, identical all over the cosmos, and not dotted with high-entropy black holes, and so forth.

This discovery soon gave way to what is known as the horizon problem. In the standard big bang model, the cosmic background radiation couldn’t possibly be the same temperature everywhere. Exact thermal equilibrium between regions of space that had always been separate made no sense. In 1979, Alan Guth tackled this inconsistency when he worked out inflationary cosmology, an exciting revision of standard big bang theory.

Einstein’s equations don’t address how the expansion of the universe began, and later cosmologists followed his lead by taking the expansion as an unexplained given. Guth’s theory states that the universe existed before the bang, and that it was only the action of the repulsive gravitational force that caused the universe to explode outward, which triggered a huge burst of accelerated expansion. After this event, the standard bang theory follows as before. The difference is that Guth’s inflationary cosmology describes the big bang as a major event that affected the universe—not the event that created it.

If the universe existed before the bang, different regions of space have had ample time to interact and adjust their temperatures to match (the way that two rooms of a house will eventually become the same temperature if the doors connecting them are open long enough). In the very beginning of time, space expanded slowly enough for a uniform temperature to be established, and only then did the massive bang accelerate the expansion. During the inflationary period, the universe was dominated by a cosmological constant that later decayed to form the matter and radiation filling the universe today.

This model does much to explain why we can only see three of the ten dimensions string theorists believe exist. String theory reduces the lower limit of the original (that is, pre-bang) universe’s size to about Planck length. Vafa and Brandenberger argue that at about Planck time, when the inflationary bang occurred, three of the tightly curled-up dimensions (in the beginning, all are curled) were chosen at random. These three then expanded rapidly to extended spatial dimensions. String theory, Veneziano has concluded, is in no way inconsistent with inflationary cosmology.

After sketching out a few alternative hypotheses about the pre-big-bang universe, Greene tries to explain M-theory’s treatment of the always-troublesome subject. M-theory, like string theory, conceives of gravity as merging with the other three forces, and does not require extreme states of infinite compression and energy to enter the scenario.

Greene discusses physicists’ speculations about the possible existence of a larger multiverse. If a larger multiverse exists, our universe would be simply an island randomly selected for inflationary expansion. Other universes may undergo periods of expansion at other times and emerge with entirely different laws of physics: different particle properties, numbers of dimensions, and so forth. But our universe, for whatever reason, possesses the specific properties that make life possible. The universe has the properties we observe because, were the properties any different, we would not be here to observe the change. This is called the weak anthropic principle.

Lee Smolin, who was interested in the similarities between the big bang and the center of black holes, has argued that every black hole contains the seed for an all-new universe. This would mean that universes capable of forming black holes have greater reproductive mechanisms and thereby come to dominate the multitude of universe within the multiverse.