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The Elegant Universe Chapter Summaries Editor: Ross Cheung, minor corrections by Wing Ning Yung Chapter 1, part 1: “String Theory: The Basic Idea" Kumar Jeev The first chapter "Tied up with Strings" starts off with a description of the inherent antagonism between the two "foundational pillars of modern physics - quantum mechanics and theory of general relativity. Quantum mechanics provides the set of laws explaining phenomena at the small scale while general relativity does the same for the large scale. Unfortunately, these two theories are incompatible. This provides the motivation for further research to search for some kind of unification of the two using the string theory. The author then begins a brief overview of matter at the microscopic level and describes the search of the Greek "atom" (last indivisible part of all matter). In this process the author brings us up-to-date on the most recently discovered three families of fundamental particles. To complete the overview the author finally describes the four fundamental forces of nature - gravity, electromagnetism, strong, and weak and the particles associated with them. Chapter 1, part 2 Ross Cheung This section introduces the concept of string theory, the theory that all electrons and quarks are composed of a "string," a one-dimensional infinitesimally thin loop. String theory suggests that all matter and all forces are determined solely by the oscillations and vibrations of this string. For this reason, many scientists, with varying degrees of skepticism, see string theory as perhaps the unified field theory, or "Theory of Everything", that has been the Holy Grail of physics for much of the 20th century. This is, however, a controversial school of thought. Staunch reductionists claim that virtually everything in the world can be explained in terms of the properties of this "string theory", an outlook that many are uncomfortable with. Opponents of reductionism claim that new developments such as chaos theory state that at every increasing level of complexity, new laws come into play, which cannot be reduced to a simpler set of laws. The author suggests that the new laws that are formed with each increasing level of complexity aren't necessarily independent of each other, but can be reduced to a simpler level; they exist because to reduce all the laws to a basic set of principles would be virtually impossible due to the level of complexity that would result. Finally this section ends with the current state of string theory. The problem with the theory is that due to technological limitations, the theory hasn't yet been confirmed with any rigorous experiments and has yet to be accepted by the scientific community. The theory has revealed numerous insights into the nature of our universe, and presents solutions to certain conflicts which have troubled physicists for some time. However, it can at best be viewed as a work in progress. Chapter 2, Part 1: “Space, Time, and the Eye of the Beholder” Kancy Lee In the mid 1800s, after the successful work of the physicists Michael Faraday and James Maxwell that united electricity and magnetism in the framework of the electromagnetic field, the paradox about light greatly troubled Einstein and many other physicists. Maxwell’s theory showed that visible light is a kind of electromagnetic wave that always travels at light speed, never stopping or slowing down. However, according to Newton’s law of motion, light will appear stationary if we travel at the same speed. Not until 1905 did Einstein resolve the conflict through his special theory of relativity, introducing new conceptions of space and time. Special relativity claims that observers in relative motion will have different perceptions of distance and time. In other words, individuals who are moving with respect to each other will not agree on their observations of either space or time. The effects of special relativity depend on how fast one moves: as the relative velocity of individuals gets larger, the differences in measurements of time and distance become increasingly apparent. These phenomena are called “time dilation” and “Lorentz contraction”. The principle of relativity states that all constant-velocity, force-free motion is relative, and that the laws of physics are absolutely identical for all observers undergoing constant-velocity motion. Chapter 2, Part 2 Daniel Ho Time is a very abstract concept, but for simplicity, we may consider it as something measured by clocks. However, motion can affect the passage of time, i.e. the clocks, which even include the biological clocks in our bodies, making them run slower or faster due to their motions. Not only time but space is also affected by motion. For example, the length of a moving car measured by a stationary observer standing on the ground is shorter than its original length. However, all these effects are hard to detect in our daily lives as the speeds we usually encounter are very slow compared to light speed. Einstein interprets all these phenomena by just considering space and time as different dimensions of spacetime. He thinks that all objects are always moving in spacetime at light speed, and the phenomena of special relativity are due to the sharing of motion between the space dimension and the time dimension when an object is moving. Not only are space and time interwoven and twisted with each other, another example is energy and mass, which leads to Einstein’s most famous equation: E = mc2. Energy and mass are not independent. They are convertible to each other, just like coins and banknotes. Lastly, nothing can travel as fast as light. The reason is that the faster an object is travelling, the greater its energy will be. According to Einstein’s most famous equation, the object mass will increase as the energy increases. As a result, it is harder and harder to push the object. In fact, if the object’s speed is at the light speed, its energy is infinite, which is certainly impossible to achieve. This is why nothing can travel as fast as light. Chapter 3: “Of Warps and Ripples” Ting Liao In chapter 2, the author explained how special relativity resolved the conflict between the classical intuition about motion and the constancy of the speed of light. In chapter 3, he turns to the conflict resulting from special relativity and the Newtonian gravitational law. According to the Newtonian gravitational law, the gravitational force depends only on the masses of the objects and the distance between them. That means, if I move one of the objects, the other object will feel the change in gravitational force instantaneously since the distance is changed. However, according to special relativity, no information can travel faster than light, so the object cannot feel the change in the force instantaneously. Something is inconsistent. Einstein, who had a lot of faith in special relativity, decided to find out what gravity is. Gravity had been a black box to human beings before Einstein. No one knew how it worked but only what it did, and this made it very hard for Einstein to start. On a day in 1907, while Einstein was working on this issue in the patent office in Bern, Switzerland, he had an idea about the link between gravity and acceleration. The author used an example to explain the idea. If we want to send something to outer space without changing the down-pull force that it is feeling, we can use the acceleration of the rocket to make the object feel the force that is equal to the gravitational pull when it is on the ground. This idea led Einstein to think of gravity and acceleration as equivalent. Therefore, to work out what gravity is really doing, he started out with acceleration. This was incredibly difficult even for one of the brightest minds in history. Using special relativity and the link between gravity and accelerated motion, Einstein made another breakthrough. To explain this, the author used an example of motion in a circle. In a circular motion, the speed is constant and the acceleration only affects the direction of the motion. If a person is inside the object that is rotating, say a tornado, and he tries to measure the ratio of the circumference to the radius, what would he find? For us, the bystanders, we will find a ratio of 2*pi. However, due to special relativity, the person’s ruler will be contracted (shorter) due to Lorentz transformation since his ruler will be in the direction of the velocity. To him, the ruler is still the same length because they are in the same reference frame. Therefore, the circumference length he measures will be longer. On the other hand, the radius is perpendicular to the velocity, so there is no contraction and the radius measured will be the same as what we measure. As a result, the ratio will not agree. Why is the ratio different? To answer this question, Einstein proposed that space is not flat when there is acceleration. In other words, what gravity does is actually curve the space. When things are traveling, they travel in the shortest path. According to general relativity, the “pull” is just the curvature leading objects to accelerate. The image is very hard for us to imagine since we live in a three dimensional of apparently non-curving space, curvature of space is hard to visualize. Besides, the time is also curved since time is just another dimension, but this is so complex to understand so the author did not spend much time on this. For some time there was no confirmation of general relativity. Finally, using the eclipse in 1919, scientists showed that general relativity predicted the angle of the incoming light precisely. Since then, different experiments have successfully confirmed general relativity. Yet, there is still a problem. The original form of general relativity predicts the universe to be expanding, thus Einstein added a cosmological constant to make it satisfy a static universe. When Hubble found proofs that the universe is in fact expanding, Einstein removed that term, (although we now believe that that term might actually be needed to describe the acceleration in the expansion of the universe). General relativity deals with objects with great mass, such as stars and black hole. The problem is that black holes can be really small but really massive, and we have to use quantum mechanics and general relativity to explain them at the same time. There is where the conflict between the two theorems arises. Chapter 4, Part I: Microscopic Weirdness The chapter "Microscopic Weirdness" is an introduction to quantum mechanics. The 'weirdness' of the quantum phenomena is a highlight of the H-bar where the fictional trans-solar system explorers Gracie and George share a drink in the introduction of this chapter. The author then quickly moves further to establish quantum mechanics as the "conceptual framework" to understand the microscopic universe. He introduces the central problem of nineteenth century physics, that of the lack of understanding of the dual nature of electromagnetic radiation. Planck's hypothesis, about the lumpy nature of energy and the existence of Planck's constant as the proportionality constant between frequency of a wave and its energy, bridges the gap in this lack of understanding. Plank's theory generated a new question about the reason of the existence of these energy lumps. This was answered by Einstein in his paper on the Photoelectric Effect who proposed the existence of photons, the light quanta. The author then describes Young's double slit experiment which arguably proves the wave-nature of light. This in turn brings the "wave-particle duality" to the forefront which sets the need for a theoretical understanding of the 'weird' micro-cosmos. Chapter 4, Part II: Microscopic Weirdness Kancy Lee In 1926 German physicist Max Born suggested that the wave nature of matter can only be interpreted in a probabilistic manner: we can only predict that the location of an electron may be found with a non-negligible probability, but we cannot predict the exact location of that electron. Although Schrödinger’s equation, that governs the shape and the evolution of probability waves, can make accurate predictions which agree with experimental results, many still find this idea unacceptable and there is no consensus on what it means to have probability waves. Richard Feynman was one of the greatest theoretical physicists. He fully accepted the probabilistic interpretation, but argued that any individual electron traveling from one location to another actually traverses every possible trajectory simultaneously. By assigning a particular number to each of these trajectories, Feynman’s formulation shows that the combined average of these paths can give results that agree with those of Schrödinger’s equation and experiments. Therefore, Feynman’s perspective, which does not require the probability wave, has provided a different approach to quantum mechanics. The hallmark feature that fundamentally differentiates quantum from classical reasoning is the uncertainty principle. Discovered by German physicist Werner Heisenberg in 1927, the uncertainty principle states that there is always some disruption to the electron’s velocity through our measurement of its position. Heisenberg found that the mathematical relationship between the two is inversely proportional to each other. That is, the greater precision in a position measurement entails greater imprecision in a velocity measurement, and vice versa. Likewise, he also found that the increasing precision of energy measurements require longer durations to carry them out. Hence, according to Heisenberg’s uncertainty principle, quantum mechanics allows a particle to borrow energy within a certain time frame. Chapter 5 Daniel Ho The heart of quantum mechanics is the uncertainty principle, and it reveals the uncertainty and frenzy in the quantum world. The Schrodinger Equation is also another important description of the quantum world. However, physicists found that it is only an approximation, as it does not include special relativity. As a result, physicists started to seek a new theory that includes both of them. They created the quantum field theory, which helps to describe three fundamental interactions. Then they generalized all they know about these three interactions into a theory called the standard model. According to the standard model, there are messenger particles responsible for different types of interactions. They help to transmit the message of how different particles should interact with others. There is still one fundamental interaction that has not been described with its own quantum field theory: gravitation. In order to create this quantum field theory, physicists have to combine quantum mechanics and general relativity. However, physicists found that they are not compatible. The reason is that the smooth spatial geometry, which is the central principle of General Relativity, is destroyed by the violent quantum fluctuation when the spacetime is examined in a very small scale. Mathematically, the combination of equations of General Relativity and quantum mechanics yields a non-physical solution: infinity. As a result, a new theory found to combine them is the superstring theory. Chapter 6 : “Nothing but Music : The Essentials of Superstring Theory” Ting Liao The chapter started off explaining why we need string theory. The standard model (before string theory) views the elementary constituents of the universe as point-like ingredients with no internal structure. It can explain almost everything, except gravity. Moreover, attempts to incorporate gravity into its quantum-mechanical framework have failed due to the violent fluctuations in the spatial fabric that appear at ultramicroscopic distances (shorter than the Planck length). With these problems present, physicists started to search for a deeper understanding of nature. Michael Green and John Schwarz provided the first piece of convincing evidence that superstring theory might provide this understanding. The author then turned to talk about the history of string theory. Physicists first proposed the string theory because it can explain why the nuclear interactions are described by the Euler beta-function. However, the string model led to a number of wrong predictions and was abandoned by most physicists in the 1970s. Yet a few physicists kept working on it. Their work was ignored by most people, partly due to the conflicts between string theory and other model, such as quantum mechanics. Such was the case until 1984, when Green and Schwarz found that the subtle quantum conflict afflicting string theory could be resolved. They also showed that the resulting theory had sufficient breadth to encompass all of the four forces and all of matter. At that point, physicists around the world all tried to work on string theory, a theory that appeared to be the theory of everything. The period from 1984 to 1986 is known as the first superstring revolution. There were great developments on string theory in those 3 years. Unfortunately, the mathematics for string theory is so complex that everyone got stuck at some point thus many decided to give up. Finally in 1995, Edward Witten announced a plan for taking the next step. The is the beginning of the second superstring revolution. We used to have no idea why particles have the properties they have. String theory offers the explanation that these properties are the allowed vibrational patterns of the strings. With the calculation on the force of gravity, physicists found that the tension of the string is in the order of 10^39 tons. That led to three consequences. Firstly, the huge string tension causes the loops of string theory to contract to a minuscule size. Secondly, the typical energy of a vibrating loop in string theory is extremely high. With that said, how do the strings build up the particles with low energy? The author explained due to quantum mechanics, there is a quantum fluctuation that cancels the energy and give rise to the low energy particles. The third consequence is that the strings can execute an infinite number of different vibrational patterns. However, all but a few of these vibrational patterns will correspond to extremely heavy particles. With our technology, it is very hard to search for these heavier particles. How, then, does the string theory solve the conflict between general relativity and quantum mechanics? Basically string theory softens the violent quantum undulations by “smearing” out the short-distance properties of space. The author gave a rough and a precise answer about what this means. The basic idea is the same. To probe something, the “object” we use as a probe cannot be larger than the feature we are looking at. The most appropriate measure of a particle’s probing sensitivity is its quantum wavelength. In 1988, David Gross and his student Paul Mende showed that when quantum mechanics is taken into account, continually increasing the energy of a string does not continually increase its ability to probe finer structures, in direct contrast with what happens for a point particle. The upshot is that no matter how hard you try, the extended nature of a string prevents you from using it to probe phenomena on sub-Planck-length distances. The conflict between general relativity and quantum mechanics arises when looking at sub-Planck-length distances. Now that this tiny length is smeared out, the fluctuation according to quantum mechanics is smoothed out just enough to cure the incompatibility between the two theories. Chapter 7 Kumar Jeev The chapter "The Super in Superstrings" is concerned primarily with the principle of supersymmetry in string theory. The author starts off with an introduction of the various symmetries that exist in nature and then talks about the most recently discovered symmetry of nature - that related to the spin of a particle (Supersymmetry). Supersymmetry however required the existence of superpartners, whose existence has not yet been experimentally verified. Before string theory came in, the case for supersymmetry was weak. It was supported merely by the aesthetics of symmetry, ease of fitting gravity into the quantum model, and its support of the grand unification of nature's forces. These justifications although were significant were not rigorous. It was only after the advent of string theory that the profound concept of super symmetry got integrated into physics. However, this integration created problems for string theory because it resulted in the creation of five different and totally consistent string theories. The author ends the chapter with a note that these seemingly different theories have now been shown to be different ways of describing "the one and the same overarching theory". Chapter 8: More Dimensions Than Meet the Eye Kancy Lee Perceptions are based on experiences. Beliefs and expectations from common experiences are the most difficult to challenge. In 1919, a Polish mathematician Theodor Kaluza had the temerity to take the challenge. He suggested that the universe might have more than three spatial dimensions. Later in 1926, a Swedish mathematician Oskar Klein refined Kaluza’s work, and proposed that the universe may have additional spatial dimensions that are tightly curled up into tiny spaces which even the most refined experimental equipment cannot detect. Just like a garden hose which may look like a thin, one dimensional line from a substantial distance, we can see its curled-up dimension when we move closer. Thus the universe should have at least four spatial dimensions: three large, extended dimensions, and one small, circular dimension. Based on the possibility of having an extra spatial dimension, Kaluza argued that both gravity and electromagnetism are associated with ripples in the fabric of space. Gravity is carried by the familiar large and extended dimensions, while electromagnetic force is carried by the new curled-up dimension. Although Kaluza’s equations, which resulted from analyzing the general relativity with the extra dimensions, were strikingly similar to those used to describe electromagnetic force, his theory was unable to incorporate all features of forces and matter. But nonetheless, Kaluza’s and Klein’s reasoning had indicated that the universe may very well have more dimensions than meet the eye, and hence provided a compelling framework to relate Einstein’s general relativity and Maxwell’s electromagnetic theory. Chapter 9: “The Smoking Gun: Experimental Signatures” Daniel Ho However elegant the string theory is, it is not easy for physicists to accept it without experimental verification of its predictions. Also, it should provide an explanation of phenomena not understood by any other current theory. In the past, many physicists did not support the string theory just because it is not easy to find experimental results of verification. As a result, they debated whether they should do any further research on string theory. Strings theorists believe the lack of verifying experimental results is due to a lack of sufficient technology. This does not mean that string theory is fundamentally divorced from experiment. However, string theory can be verified in an indirect manner, using things like superparticles, fractional charged particles, etc. In addition, string theory gives the framework to explain the properties of the elementary particles and messenger particles. Nevertheless, the story is not that simple. There are many Calabi-Yau shapes, which is a very important concept in string theory, making it possible for us to explain our world. The equations of string theory are only an approximation and they are not strong enough to tell which shapes are most suitable. Furthermore, string theory may be able to explain the data and properties of neutrinos, especially if neutrinos are found to be of non-zero mass, since according to standard model, neutrinos have zero mass. String theory may also be able to find the possible candidates for dark matter, which is important in cosmology. Finally, compelling evidence may come if string theory can help improve the mismatch between the calculated value and the observed value of the cosmological constant. Chapter 10: “quantum geometry” Ting Liao Einstein’s General Relativity changed the world’s view about gravity, but it was Riemann who broke the chains of flat-space Euclidean thought and paved the way for a democratic mathematical treatment of geometry on all varieties of curved surfaces. It was Riemann’s insights that provide the mathematics for quantitatively analyzing warped spaces. While in string theory we need to modify general relativity in small distances in the Planck’s scale, we need another geometry to describe the physics, and that is what we call quantum geometry. In general relativity, the curvature of spacetime reflects the distorted distance relations between its points. By making the object smaller and smaller, the physics and the mathematics align ever more precisely as we get closer and closer to physically realizing the abstract mathematics of a point. However, in string theory, we cannot get any smaller than a string, so the geometry is modified on ultramicroscopic scales. According to the big bang model of cosmology, the universe violently emerged from a singular cosmic explosion. Today, the “debris” from the explosion is still streaming outward. If the total mass in the universe is great enough, the gravity may be so strong that all the mass collapses to one singular point again according to the theory of general relativity. However, when the distance scales involved are around the Planck length or less, quantum mechanics invalidates the equations of general relativity. String theory again gives a limit on how small the universe can be. We now turn to how string theory changes the picture. For simplicity, we think about the problem in a garden-hose universe. The main difference between a garden-hose universe and a flat universe is that string can be wrapped around the universe in a garden-hose universe. That implies that the energy of a string in the garden-hose universe comes from two sources: vibrational motion and winding energy. Vibrational motion is further split into ordinary vibrations (usual oscillations) and uniform vibrations (the overall motion of string as it slides from one position to another without changing its shape). Uniform vibrational excitations of a string have energies that are inversely proportional to the radius of the circular dimension according to quantum mechanics. On the other hand, the winding mode energies are directly proportional to the radius. So when a universe is becoming smaller, the uniform vibrational energy is increasing and the winding mode energies are decreasing. Since we care only about the total energy of a string, as long as the sum of the two energies is the same, it is the exact same configuration. Also if we exchange the energy between the vibrational energy and the winding mode energy we can get the same state. But the difference in the distribution of energy means the universe has a different size. What that means is that for any size of the universe, there is a corresponding size that gives the exact same physics, so when the universe is crushing down, it can be looked at as a collapsing to the planck’s length and then expanding again. Before we get any further, we have to understand what distance is. There are two different yet related operational definitions of distance in string theory. Each lays out a distinct experimental procedure for measuring distance and is based, roughly speaking, on the simple principle that if a probe travels at a fixed and known speed then we can measure a given distance by determining how long the probe takes to traverse it. The difference is in the choice of probe used. The first definition uses strings that are not wound around a circular dimension where the second uses strings that are. Unwound strings can move around freely and probe the full circumference of the circle, a length proportional to R. By the uncertainty principle, their energies are proportional to 1/R. On the other hand, wound strings have minimum energy proportional to R, and are sensitive to the reciprocal of this value, 1/R. Therefore, using different probes to measure distance can result in different answers. This property extends to all measurements of lengths and distances. The results obtained by wound and unwound string probes will be inversely related to one another, and they are equivalent. However, no one knows for sure if these conclusions hold if the spatial dimensions are not circular in shape. Now that we have seen radius of R and 1/R are indistinguishable in string theory, we want to know if it is possible to have equivalent strings with different shapes. In 1988, physicists, based upon aesthetic arguments rooted in considerations of symmetry, made the audacious suggestion that it might be possible. A key fact is that the number of families of particles arising from string vibrations is sensitive only to the total number of holes, not to the number of holes of each particular dimension. So we can have calabi-yau spaces with different configurations but that give rise to same number of families. Although this is just one property, it shows how the hypothesis can be true. Later, people used orbifolding to look at the changes in calabi-yau shapes. They found that if particular groups of points were glued together in just the right way, the Calabi-Yau shape produced differed from the initial one in a startling manner. The number of odd dimensional holes and even dimensional holes were exchanged. The number of holes is thus the same, but the even-odd interchanges means the shapes and fundamental geometrical structures are quite different. After that, people started asking about if other physical properties will be the same in these different shapes. The answer was very likely affirmative. This symmetry between different shapes, called mirror symmetry, helps in finding certain aspects of calabi-yau spaces. For a problem in one space, there may be an easy way of solving it in the symmetric space. Using this method, physicists have been able to solve difficult mathematical problems. String theory not only provides a unifying framework for physics, it may well forge an equally deep union with mathematics as well. Chapter 11: Tearing the Fabric of Space Kumar Jeev This chapter is the success story of string theory over the other existing theories in discovering the nature of our universe. Although the tearing of the spatial fabric is impossible according to Einstien's general relativity, this chapter talks about how string theory shows the converse. Yau and Tian's seminal work on constructing new Calabi-Yau spaces by "tearing" and "stitching" existing Calabi-Yau spaces opened the avenue of research in relating flop-transitions to the "real" spatial fabric. The author in the sections that follow writes about the events of the fall of 1992 when at the Institute of Advanced Study, Green, Aspinwall, and Morrison on one side and Witten on the other, showed independently that the space in fact can undergo tearing and probably is undergoing an elongated rupture. While Green and Aspinwall's approach involved the mathematics behind mirror symmetry of Calabi-Yau spaces, Witten's approach handled the problem from a microscopic point of view. Chapter 12: “Beyond Strings: In Search of M-Theory” Kancy Lee Although string theory appears to be able to provide a unique picture of the universe in the late 1980s, there are actually five different versions of string theory: Type I, Type IIA, Type IIB, Heterotic-O, and Heterotic-E. Many string theorists had been trying to combine all five distinct string theories into a single, all-encompassing framework. During a conference at the University of Southern California in March 1995, Edward Witten announced a new strategy for transcending the perturbative understanding of string theory, introducing a new approach that is rooted in the power of symmetry and the concept of duality. He suggested that the five string theories, although apparently different in their basic construction, are just different ways of describing the same underlying physics. Witten was the first to come upon a unique framework, which has eleven dimensions for a more satisfying synthesis of the theory, called the M-theory. Even though many issues remain unresolved, and we have only a scant understanding of Witten’s discovery, the M-theory has provided a substrate for a far grander unifying framework of string theory and has opened our eyes to a whole slew of different spatial dimensions. Chapter 13 Daniel Ho Both black holes and elementary particles can be described by their mass, force, charges and spins. It seems that a black hole is just a huge elementary particle. In order to understand this, string theory is needed. This chapter is about the development of string theory and how it eventually explains this relationship. It begins with a problem about the collapse of a three-dimensional sphere surrounded by a Calabi-Yau shape’s fabric. Not until 1995 did the string theorists think this would not lead to catastrophes. At that time, Strominger showed that the three branes, which were discovered just before, provide a shield that completely cancels out the cataclysmic effects arising from the collapse of the three-dimensional sphere. Then, other physicists started to study the collapse of this sphere and they found out that it would tear and subsequently repair itself, becoming a two dimensional sphere. This means one Calabi-Yau shape can transform into another shape. This has something to do with black holes and elementary particles. String theorists found out that the gravitational field of a black hole is similar to that set up by the three-brane smeared around the three dimensional sphere. They also found out that the mass of a black hole, which is just the mass of the three-brane, is proportional to the volume of the three dimensional sphere it wraps. To conclude, a massive black hole becomes ever lighter, as the three dimensional sphere is collapsing, until it is massless and then it transmutes into a massless particle. This phenomenon shows that black holes and elementary particles are just different phases of the same underlying stringy material. Physicists also asked what the entropy of a black hole is. In 1970, Jacob Bekenstrin, then a graduate student in Princeton, thought that the total area of the event horizon of a black hole provides a precise measurement of the entropy. His suggestion was not accepted until Stephen Hawking found that black holes emit radiation quantum mechanically. From their prediction, they found that the entropy of a black hole is extremely large. This confused physicists very much as they thought that black holes are very ordered object and so the entropy should be small. However, string theorists solved this problem. They set up a theoretical black hole, which is based on the string theory, and they calculate the entropy of this black hole. It turns out the result agrees perfectly with Stephen Hawking and Bekenstein’s result. Chapter 14 Ting Liao Humans have been asking questions about the origin of the universe for centuries, and we have now come to a point where a framework is emerging for answering some of these questions scientifically. The currently accepted model is the big bang model. The model, which is usually referred as the standard model of cosmology, says that the universe was initially in an extreme condition with infinite mass, energy and density. To understand the infinitely massive universe with an infinitely small volume, we had to use both the theory of general relativity and quantum mechanics. String theory changes our view on the standard model by changing our views on the two physics. In the standard model, the universe erupted from an enormously energetic, singular event, which spewed forth all of space and all of matter. As time passed, the universe expanded and cooled, and particles started to come together and formed bigger particles or substances as we see today. When charged particles (electron and proton) merged, they formed a neutral particle. Since neutral particles do not affect light, which is basically electromagnetic wave, some time after the big bang, the photons were free to travel through the universe. The detections of these photons have given us evidences supporting the standard model. Furthermore, the predictions about the relative abundance of the light elements produced during the period of primordial nucleosynthesis have largely agreed with experiments. However, both these results correspond to conditions some time (about a hundredth of a second) after the big bang. Physicists believe that between the Planck time (10^-43 seconds) and a hundredth of a second after the big bang, the universe passed through at least two phase transitions. At temperature above 10^28 Kelvin, the three nongravitational forces appeared as one, as symmetric as they could possibly be. But as the temperature dropped below that value, the three forces broke out from their common union in different ways. At this point the weak and electromagnetic forces were still deeply interwoven. When the universe reached 10^15 Kelvin, these two forces also broke out from the union and became more different as the universe cooled down. This is why the forces are so different to us nowadays. Now we turn to the horizon problem. The problem is that when we look at the cosmic background radiation, no matter which direction we look at, we get the same result. However, in the standard model, the separation between the two ends of the universe has always been so far that, information has never had enough time to travel from one end to another. Therefore they could not have had exchanged information any time in the past. How could they be the same without any connection? The explanation is that at around 10^-36 seconds after the big bang, the universe underwent a process called the inflation. During that time, the universe expands much quicker than normal. What that means is that the size of the universe before the inflation is much smaller than we expected, so there was enough time for information to travel allover the universe. We have just talked about the evolution of the universe after Planck’s time, now we should take a look at what was going on before Planck’s time. As mentioned, since the size of the universe was so small yet so massive, we have to merge general relativity and quantum mechanics, and this is where string theory comes into play. There are three essential ways in which string theory modifies the standard cosmological model. First, string theory implies that the universe has what amounts to a smallest possible size, instead of a infinitely tiny size. Second, string theory has a small-radius/large-radius duality. What that means is when the universe collapses to the Planck’s length, any further collapse will be equivalent to an expansion. In other words, we can say that the universe is in a cycle of expanding and collapsing to the Planck’s size. Therefore, some physicists suggest that at the beginning, the universe was a nugget with Planck’s length in every dimension. Finally, string theory has more than four space-time dimensions, and from a cosmological standpoint, we must address the evolution of them all. The first question comes into mind is that why do only three of the spatial dimensions expand. The basic answer is that in four or more space dimensions wrapped strings are less and less likely to collide. Since half of the collisions will involve string/antistring pairs, leading to annihilations that continually lessen the construction, allowing these three dimensions to continue to expand. There are a lot of other different insights raised by string theory, but they’re too complicated to explain. Although string theory gives a lot of new insights, there are just too many possibilities on how the universe actually is. Right now it is too hard to tell if string theory is really the theory of everything. Nevertheless, the cosmological implications of string/M-theory will be a major field of study in this century. Without accelerators capable of producing Planck-scale energies, we will increasingly have to rely on the cosmological accelerator of the big bang, and the relics it has left for us throughout the universe, for our experimental data. Chapter 15 Ross Cheung This chapter exists to conclude and tie together many of the ideas presented in this book. To do so, it presents numerous questions about the nature of the universe. The first one asks, is string theory the ultimate fundamental principle that explains the laws of the universe, or is it just the next step, the next deepest level of understanding? In the past, whenever we believed we had figured out everything, it turned out there are more beneath it all. However he points out that all of the principles of symmetry, which are associated with all the known laws of Physics, emerge from the structure of string theory. The second question deals with the nature of space and time, and what it is. While there are numerous ways of visualizing it, the puzzling question is what is really meant by the fabric of the universe? Isaac Newton believed that the universe is constant and unchanging, and thus space and time were absolute. Gottfried Leibniz and many others argued that space and time are merely bookkeeping devices used to summarize relationships between objects and events in the universe, a viewpoint that has been supported by developments in string theory. The third question considers whether or not string theory will reformulate quantum mechanics. So far, developments in string theory have followed that of many classical principles, mainly that string are first described in classical terms, and are then converted into a quantum-mechanical framework. However, there are suggestions that this method is inherently flawed, and if the universe is governed by quantum mechanical principles, the theories to explain it should be quantum mechanical from the start. So far scientists have been able to get away with it, but as string and M-theory gets deeper, it may not be possible to continue doing so. The fourth question deals with whether or not string theory can be experimentally tested. While the theory fits with all of the other principles of Physics, it has yet to be tested experimentally because of technological limitations. Larger and larger particle accelerators are being built right now, but whether they will be sufficient remains to be seen. The fifth and final question is whether there are limits to explanation. Even if we were to have perfect understanding of string and M-theory, what if we were still unable to calculate particle masses and force strength? More importantly, would the failure of string theory reflect that those observed properties have no explanation? The answer hinges on whether string theory is ultimately the deepest level of understanding in the universe. If so, then its failure means that there are just some things in the universe that science can not explain.

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