## New Developments in String Theory

String theory since the second superstring revolution has become one of the most researched physics concepts of our time. Since a catalog of new discoveries in string theory would be extremely long and difficult, both to compile and to read, those developments that focus specifically on cosmology or that have a direct relationship to cosmology have been chosen for presentation here. See the bottom of this page for suggested readings which will provide additional information.

## Mirror Symmetry

Although a circle of radius R is obviously drastically different from one of radius 1/R, the physics they yield is identical. This led physicists to ask the next logical question: are there geometrical forms in the universe that yield the same physics but differ in shape instead of size? Based on symmetry principles, some physicists argued that two completely distinct Calabi-Yau shapes chosen as the curled-up dimensions might give rise to the same physical properties. Later, the technique of orbifolding to produce the Calabi-Yaus was introduced. Orbifolding is a procedure by which points on a Calabi-Yau are systematically connected so that a new Calabi-Yau with the same number of holes as the original is produced. After research into the procedure, it was found that orbifolding in a particular manner yielded an interesting result: the number of odd-dimensional holes in the revised shape (recall that Calabi-Yau holes can have many dimensions) equaled the number of even-dimensional holes in the original and vice versa. The result was that, even though the shape of the Calabi-Yau had been fundamentally changed, it would yield the same number of particle families - one step toward the idea of separate Calabi-Yaus generating identical physics. When the two Calabi-Yaus' physical implications were analyzed, it was found that they produced not only the same number of families, but also the same physical properties. Two Calabi-Yaus differing in the way described above but yielding the same physics are called mirror manifolds, and this property of string theory is called mirror symmetry, although the holes are not reflected in the ordinary manner.

## Applications of Mirror Symmetry

Mirror symmetry in essence claims that two seemingly unrelated Calabi-Yau spaces with their holes correlated in the above manner generate the exact same physics. This provides an important mathematical tool for those cases when the detailed implied properties of a Calabi-Yau space's associated universe are being calculated and an incredible mathematical difficulty crops up. Mirror symmetry allows the difficult calculation to be rephrased in terms of the Calabi-Yau's mirror partner. The resulting physics will be the same, but the difficulty of the calculation will probably have been greatly reduced.

## Rips and Tears in the Spatial Fabric

Stretching a common analog of spacetime, a rubber sheet, will eventually break it. Physicists have long wondered if this is also true of the fabric of spacetime itself. According to general relativity, which relies on smoothly curving geometrical spacetime, the spatial fabric cannot tear or rip. However, the violent fluctuations of quantum mechanics seem to imply the possibility of minute breaks in the spatial fabric. The most often-debated such break is the wormhole of the type seen by Star Trek fans on Deep Space Nine. Wormholes are simply tunnels that constitute shortcuts from one area of space to another. In the diagram, the traditional example of a U-shaped universe is presented. The only way to get from one place to another - in the image, from Point A to Point B - is to proceed along a straight path through the recognized cosmos (red path). If a wormhole (dark ellipses and white tunnel) is present, a traveler may utilize its shortcut to bypass the long, normal route and reach the Point B much faster (blue line). However, if wormholes do exist, it is unclear whether they will exist in this macroscopic form that facilitates traveling or simply in microscopic form as part of quantum foam.

Black holes are another example of significantly stretching the fabric of space. The fabric of space certainly seems to be punctured at the center of a black hole. Some suggest that this type of singularity can occur, but only when isolated from the normal rest of the universe by the event horizon, from which nothing can escape. A recent theory proposed by Shing-Tung Yau and his student Gang Tian stated that Calabi-Yau shapes could be transformed into others by breaching their surfaces, then reconnecting them according to a mathematical formula. They utilized a two-dimensional sphere, like the surface of a ball, and shrank it to a single point. They then postulated slightly opening the shape at that point and adding another two-dimensional sphere to replace the first, which would then reexpand. Since this procedure results in the original sphere being "flopped," the transition is dubbed a flop transition. The Calabi-Yau thus formed is topologically distinct from the original; the two cannot be turned into one another without tearing or ripping.

Physicists Brian Greene, David Morrison, and Paul Aspinwall set out to prove that the above method for tearing space could indeed occur in nature. They decided to employ mirror symmetry to analyze the physics of the original Calabi-Yau after the tear and compare the results to those derived from its mirror. If the results were identical, it would prove that flop transitions could occur in nature without causing a breakdown of the spatial fabric. After many months of intense work, they proved that it was possible: flop transitions did not destroy the mirror relationship and therefore, such events were theoretically possible.

Another physicist, Edward Witten, worked on the same problem using a very different approach. He showed that strings, which have spatial extant, can travel while encircling the tear as it takes place. This, in effect, cancels out the potentially disastrous consequences of an actual rip in space. This conclusion relies on the fact that the two-dimensional worldsheet a string "sweeps out" as it moves essentially shields the rest of the universe from the potential effects. Moreover, there is always a string available to provide the shield - according to Feynman's sum-over-paths method of quantum mechanics calculations, all objects travel from one place to another by traveling along all possible trajectories. This ensures that a infinite number of strings will be passing by the tear when it occurs, thus protecting the universe from the effects of the singularity.

## Black Holes, String Theory, and Elementary Particles

Black holes and elementary particles might seem as unrelated as two objects can get: one is the enormous result when an incredibly massive star collapses due to its own immense gravity, while the other is an infinitesimally tiny constituent of all matter in the universe. However, they do have many similar features. The only attributes that distinguish one black hole from another are its mass, force charges, and rate of spin. These are exactly the three characteristics that distinguish one elementary particle from another. This has long intrigued physicists, but the lack of a way to utilize both general relativity and quantum mechanics at the same time on the same object (black holes are massive enough to require relativity and small enough to require quantum mechanics) has been a setback in research of this type. String theory, being a formulation combining relativity and quantum mechanics, may offer a way to proceed along this line of inquiry.

String theory does, in fact, give a persuasive connection between black holes and elementary particles. The idea began with an older question physicists had discussed since the 1980s. Within most Calabi-Yau shapes, there can usually be two types of spheres embedded within them: two-dimensional spheres, like the surface of a ball and the same as those used for flop transitions; and three-dimensional spheres, like the surface of a four-dimensional ball. Study of string theory revealed that it was possible for three-dimensional spheres to shrink in volume until they reached incredibly tiny size. Could the spatial fabric also collapse in an analogous fashion? Theorists postulated that the answer was no, because a string cannot encircle a three-dimensional sphere any more than a point particle could have encircled the two-dimensional sphere of flop transitions. In fact, results from certain older equations seemed to imply that the infinities tamed by string theory would reemerge if such pinching occurred. However, Andrew Strominger's groundbreaking work made use of Witten's discovery that one-dimensional strings are not the only constituents of string theory to prove that these collapses can in fact occur.

He reasoned that, just as a one-dimensional string can completely surround a circle, and just as a membrane can completely surround a two-dimensional sphere, a three-brane can completely surround a three-dimensional sphere. This three-brane wrapped around the collapsing three-dimensional sphere provides a shield that separates the universe from the potentially cataclysmic effect of the singularity. Although he showed that three-branes protect the universe from the effects of a sphere pinching to a point, he did not invoke the tearing of the spatial fabric or its repair through sphere expansion.

This is where Greene and Morrison came in again. They visualized that the sphere collapse described above could be subsequently repaired by the reexpansion of a new sphere with one less dimension than the original - a procedure called a conifold transition. Their work succeeded, but the result was a Calabi-Yau space that differed from the original in a much more drastic way that with simple flop transitions. They had proved that one Calabi-Yau could be transformed into a completely different one without causing physical calamity.

## The Implications of Conifold Transitions

Physicists found that a three-brane wrapped around a three-dimensional sphere will result in a gravitational field bearing the appearance of an extremal black hole, or one that has the minimum mass consistent with its force charges. Additionally, the mass of the three-brane is the mass of the black hole and is directly proportional to the volume of the sphere. Therefore, a sphere that collapses to a point as described above appears to us as a massless black hole, which will return to the discussion later.

Another important implication of conifold transitions is their effect on the number of holes in the Calabi-Yau space. When a pinched-point three-dimensional sphere is replaced by a two-dimensional sphere, the number of holes in the Calabi-Yau - and hence the number of low-mass vibrational patterns - increases by 1. This extra vibrational pattern turns out to be that of a massless particle into which the black hole has been transformed. Thus, as a space-tearing conifold transition occurs, the sequence of events is as follows: a massive black hole becomes increasingly lighter until it reaches the point of masslessness, at which time it transforms into a massless particle with a certain vibrational pattern. Therefore, only the precise shape of a Calabi-Yau space determines whether a particular pattern of vibration will be an elementary particle or a black hole, and black holes can undergo "phase transitions," analogous to those that change ice to water, to become a basic pattern of string vibration - a fundamental particle.

## Disordered Black Holes: Entropy and Radiation

The scientific measure of disorder is called entropy. Low entropy means a small amount of disorder and is denoted with a small number that, in essence, counts the number of ways the components of the system being measured could be rearranged without changing the overall appearance of the system. Thus, a neatly and carefully made bed has low entropy because the pillows, blankets, and sheets cannot be rearranged many ways without a change in appearance. High entropy means high disorder and is denoted with a larger number. In the above example, a messy and unmade bed could be rearranged in a number of ways without significantly affecting the overall appearance of the bed. Scientifically, entropy measure involves rearranging the quantum properties of a system while leaving its macroscopic properties intact. The second law of thermodynamics states that the total entropy of a system always increases; everything has an always-increasing state of disorder.

Physicists had long thought that black holes, with so few distinguishing properties, were some of the most organized structures in the universe. In 1970, a Princeton graduate student named Jacob Bekenstein postulated that black holes would have to have entropy in order to satisfy the second law of thermodynamics. He theorized that, in order to compensate for the entropy "lost" when disorderly matter is sucked into a black hole, the black hole itself would have to increase in entropy. He based his argument on a famous result of Stephen Hawking's - that the event horizon, or "point of no return" beyond which everything is doomed to be consumed by the black hole, always increases in total area as it devours matter. Bekenstein further proposed that the area of the event horizon provided an accurate measure of its entropy.

When Hawking made his increasing-area suggestion, he too had thought of black holes having entropy, but when he performed the calculations, discovered that a black hole with entropy would also have to have a temperature. But objects with temperatures above absolute zero are required to emit some sort of radiation - a direct conflict with the theory that nothing can escape the gravitational attraction of a black hole. Therefore, Hawking accepted the loss of entropy with black holes as an exception to the second law of thermodynamics.

In 1974, Hawking made another well-publicized discovery - black holes, contrary to previous thought, were not utterly black. The use of only general relativity to understand black holes leads to the conclusion that nothing escapes their gravity wells. However, quantum mechanics and the uncertainty principle provide a way for black holes to radiate. The uncertainty principle ensures the momentary existence of many particle/antiparticle pairs that borrow energy, then immediately annihilate. This law is true even along the event horizon of a black hole. It is possible that the gravity of the black hole might separate a pair - say, a photon pair - and one of them into its gravitational field, but give the other a boost away from the black hole. This allows black holes to radiate, thus fulfilling the requirement and allowing the physical possibility that they might have temperature and entropy, which he did in fact prove. Indeed, a black hole's temperature is just barely above absolute zero, and its entropy is truly enormous - logical considering all the matter, which naturally had disorder that had to increase when the black hole consumed it, that is contained within a black hole.

String theory was able to complete this partial result in 1996, when Andrew Strominger and Cumrun Vafa further analyzed these properties. Focusing on extremal black holes, they used BPS states to "create" a theoretical black hole by connecting branes according to mathematical rules. These "designer" black holes could then be analyzed for their entropy, the results of which agreed with Bekenstein and Hawking. This success has been considered one of the greatest achievements of string theory - it both solved a famous physics problem and used string theory to discuss at least theoretically observable phenomena.

## Black Holes as Repositories of Information

A principle called quantum determinism states that a sufficiently (nearly infinitely) powerful computer, when provided with the wave-functions of all the particles in the universe, can predict the relative probabilities of their positions and interactions for any point in the past or future. However, black holes present a twist in this reasoning - they collect vast amounts of matter whose particles of course each have wave-functions. The question is, is the information contained in these wave-functions lost forever in the black hole, or can it somehow reemerge? Before the realization of Hawking radiation, physicists theorized that the information was not lost, but it was locked away into a region of space inaccessible to anyone outside the region. However, Hawking radiation implies that a black hole's mass will slowly decrease as it radiates, thus decreasing the distance from the center of the black hole to the event horizon and re-revealing regions of space once locked away behind the event horizon. Now the question is: does the information absorbed by the black hole then reemerge? Some, including Hawking, maintain that the information is eliminated from the universe, thereby adding a new degree of randomness to the universe and destroying even quantum determinism. Others maintain that the information is not lost, but instead is contained in the branes that make up the black hole itself.

Another as yet unresolved issue of black hole physics is the state of affairs at the exact center of the huge condensed mass. Straight relativity shows that spacetime is bent into infinite curvature, thereby creating a spacetime singularity startlingly similar to that postulated by standard big bang theory. Others suggest that time itself comes to an end at the center of a black hole, which perhaps will have an effect on the eleven dimensions of string theory, since one of them is time. Still others make a closely related suggestion that black holes indeed stop time, but also create a new universe connected to ours through the singularity in which time has just begun - in effect, black holes are new big bangs that give rise to new spawn universes.

Of course, additional information to supplement that found within the string theory series is available in the other sections and pages of CosmoNet. However, the following books were used as sources for this series and would prove excellent reading to the visitor searching for additional information on string theory.

The Elegant Universe by Brian Greene

A Brief History of Time by Stephen Hawking

Black Holes and Baby Universes by Stephen Hawking

Hyperspace by Michio Kaku

Please use the selection box to progress through the string theory series.

Created by Dan Corbett, Kate Stafford, and Patrick Wright for ThinkQuest.