Shintan Yau Steve Nadis String Theory and the Hidden Dimensions of the Universe Shintan Yau Steve Nadis String Theory and the Hidden Dimensions of the Universe
This book will take you along a fascinating route to explore the hidden dimensions of space and its diversity. Written by the discoverer of Calabi-Yau space, this work describes one of the most exciting and controversial theories in modern physics.
Brian Greene, bestselling author of The Elegant Universe and The Fabric of the Cosmos Foreword Mathematics is often called the language of science, or at least the language of the natural sciences, and rightly so: laws physical world are expressed much more accurately using mathematical equations than when written or spoken in words. In addition, the idea of mathematics as a language does not allow us to properly appreciate it in all its diversity, since it creates the erroneous impression that, with the exception of minor amendments, everything truly important in mathematics has long been done.
This is actually not true. Despite the foundations created by scientists over hundreds or even thousands of years, mathematics still remains an actively developing and living science. This is by no means a static body of knowledge - however, languages also tend to change. Mathematics is a dynamic, evolving science, full of daily insights and discoveries that compete with discoveries in other fields, although, of course, they do not attract the same attention as the discovery of a new elementary particle, the discovery of a new planet, or the synthesis of a new cure for cancer. Moreover, if it were not for periodic proofs of hypotheses formulated over centuries, information about discoveries in the field of mathematics would not be covered by the press at all.
For those who appreciate the exceptional power of mathematics, it is not just a language, but an indisputable path to truth, the cornerstone on which the entire system of natural sciences rests. The strength of this discipline lies not only in the ability to explain and reproduce physical realities: For mathematicians, mathematics itself is reality.
Geometric figures and spaces, the existence of which we prove, are as real to us as the elementary particles of which, according to physics, any substance consists. We consider mathematical structures to be even more fundamental than natural particles, because they allow us to understand not only the structure of particles, but also such phenomena of the surrounding world as human facial features or the symmetry of colors. Geometers are most fascinated by the power and beauty of the abstract principles that underlie the shapes and forms of objects in the world around them.
My study of mathematics in general and my specialty - geometry - in particular has been an adventure. I still remember how I felt in my first year of graduate school, as a green youth of twenty-one, when I first heard about Einstein’s theory of relativity. I was amazed that gravitational effects and the curvature of space can be considered one and the same thing, because curved surfaces fascinated me even in my early years of studying in Hong Kong. Something about these forms attracted me on an intuitive level. I don't know why, but I couldn't stop thinking about them. Knowing that curvature underlies Einstein's theory of general relativity filled me with hope to one day contribute to our understanding of the universe.
The book in front of you tells about my research in the field of mathematics. Particular emphasis is placed on discoveries that helped scientists build a model of the Universe. It is impossible to say with certainty that all the described models will ultimately be relevant to reality. But nevertheless, the theories underlying them have an undeniable beauty.
Writing a book of this kind is, to put it mildly, a non-trivial task, especially for a person who finds it easier to communicate in the language of geometry and nonlinear differential equations, rather than in his non-native English. I was frustrated by the fact that the wonderful clarity and kind of elegance of mathematical equations is difficult, and sometimes impossible, to express in words. In the same way, it is impossible to convince people of the majesty of Everest or Niagara Falls without having pictures of them at hand.
Luckily, I received much-needed help in this aspect. Although the narration is narrated from my perspective, it is my co-author who is responsible for translating abstract and difficult-to-understand mathematical constructions into understandable (at least I hope so) text.
I dedicated the proof copy of Calabi conjecture, the book on which this edition is based, to my late father, Chen Ying Chiu, the editor and philosopher who instilled in me a respect for power. abstract thinking. I also dedicate this book to him and to my late mother Leung Yeuk Lam, who also had a great influence on my intellectual development. I would also like to pay tribute to my wife Yu-Yun, who patiently endured my inordinate (and sometimes obsessive) research and frequent work trips, and to my sons Isaac and Michael, of whom I am very proud.
I also dedicate this book to Eugenio Calabi, the creator of the above-mentioned theory, with whom I have known for almost forty years. Calabi is an extremely original mathematician with whom I have been associated for more than a quarter of a century through the class of geometric objects - Calabi-Yau manifolds, which are the main topic of this book. The Calabi-Yau conjunction has been used so often since its introduction in 1984 that I have almost become accustomed to Calabi being my name. And I would bear this name with pride.
The work I do lies at the intersection of mathematics and theoretical physics. You don't work on these things alone, so I've benefited greatly from collaborating with my friends and colleagues. I will mention only a few of the many who collaborated with me directly or inspired me in one way or another.
First of all, I would like to thank my teachers and mentors, a whole galaxy of famous scientists: S. S. Chern, Charles Morrey, Blaine Lawson, Isadore Singer, Lewis Nirenberg Nirenberg) and the already mentioned Calabi. I am happy that in 1973 Singer invited Robert Geroch to speak at the Stanford conference. It was Geroch's talk that inspired me to work with Richard Schoen on the positive energy hypothesis. I also owe my later interest in mathematical physics to Singer.
I would like to thank Stephen Hawking and Gary Gibbons for the conversations we had about general relativity during my visit to Cambridge University. From David Gross I learned about quantum field theory. I remember in 1981, when I was a professor at the Institute for Advanced Study, Freeman Dyson brought a fellow physicist who had just arrived at Princeton to my office. Newcomer Edward Witten told me about his forthcoming proof of the positive energy hypothesis, which I and a colleague had previously proven using extremely complex techniques. It was then that I was first struck by the power of Witten's mathematical calculations.
Over the years I have enjoyed collaborating with many people: the aforementioned Sean, S. Y. Cheng, Richard Hamilton, Peter Li, Bill Meeks, Leon Simon and Karen Uhlenbeck. I cannot help but mention other friends and colleagues who have contributed in various ways to this book. These are Simon Donaldson, Robert Greene, Robert Osserman, Duong Hong Phong and Hung-Hsi Wu.
I have had the good fortune to spend the last twenty years at Harvard, which is an ideal environment for interacting with both mathematicians and physicists. While working here, talking with my fellow mathematicians, I experienced many insights. Thanks for this to Joseph Bernstein, Noam Elkies, Dennis Gaitsgory, Dick Gross, Joe Harris, Heisuke Hironaka, Arthur Jaffe (who also works in physics) , David Kazdhan, Peter Kronheimer, Barry Mazur, Curtis McMullen, David Mumford, Wilfried Schmid, Yum-Tong Siew Tong Siu), Shlomo Sternberg, John Tate, Cliff Taubes, Richard Taylor, H. T. Yau and the late Raoul Bott Raoul Bott and George Mackey. And all this was against the backdrop of a memorable exchange with fellow mathematicians from the Massachusetts Institute of Technology. About physics, I had countless useful conversations with Andy Strominger and Cumrun Vafa.
Over the past ten years, I have been twice invited by Eilenberg to teach at Columbia University, where I have had fruitful interactions with other teachers, notably Dorian Goldfeld, Richard Hamilton, Duong Hong Phong, and S.W. Zhang (S. W. Zhang). I also taught at the California Institute of Technology at the invitation of Fairchild and Moores. There I learned a lot from Kip Thorne and John Schwarz.
Over the past twenty-three years, my physics-related research has received support from the US government through the National Science Foundation, the Department of Energy, and the Pentagon Office of Scientific Research. Most of my students received PhDs in physics, which is somewhat unusual for mathematicians. But it was a mutually beneficial cooperation, since they learned mathematics from me, and I learned physics from them. I am happy that many of these physics-trained students of mine have become distinguished professors in the mathematics departments at Brandeis University, Columbia University, Northwestern University, Oxford, the University of Tokyo, and other institutions. Some of them worked on Calabi-Yau manifolds and helped me write this book. Among them are Mboyo Esole, Brian Greene, Gary Horowitz, Shinobu Hosono, Tristan Hubsch, Albrecht Klemm, Bong Lian ), James Sparks, Li-Sheng Tseng, Satoshi Yamaguchi and Eric Zaslow. And finally, my former graduate students - Jun Li, Kefeng Liu, Melissa Liu, Dragon Wang and Mu-Tao Wang - also contributed your invaluable contribution to my research. I will still mention them on the pages of my book.
Shintan Yau, Cambridge, Massachusetts, March 2010 If it weren't for Henry Tai, a physicist at Cornell University (and friend of Yau), who suggested that co-authorship might lead me to interesting ideas, I probably would never have known about this project.
In this respect, as in many others, Henry was right. And I am grateful to him both for starting my unexpected journey and for helping me along the way.
As Yau often said, when you embark on a mathematical journey, you never know in advance how it will end. The same can be said for the end of the book you're working on. During our first meeting, we agreed that we needed to write a book together, but the understanding of what this book would be about did not come until some time later. You could even say that we didn't have a clear answer to this question until the book was finished.
Now, in order to eliminate any confusion, I will say a few words about the product of our cooperation. My co-author is a mathematician, whose work, in fact, formed the basis of the book. The chapters in the creation of which he took an active part are written, as a rule, in the first person. And the pronoun “I” refers to him and only him. But, despite the fact that this book is his story about himself, it is not at all an autobiography or a biography of Yau. Some of the discussion involves people Yau doesn't know (some of whom died before he was born), and some of the topics described—such as experimental physics and cosmology—are outside his area of expertise. Such sections are written in the third person and are based on various interviews and other research I have conducted.
The answer to the question of whether this book can be considered an autobiography is that while the book is undoubtedly built around Yau's work, it is assumed that the main role will be played not by himself, but by a class of geometric figures - the so-called Calabi-Yau manifold - which he helped come up with.
Broadly speaking, this book is an attempt to understand the Universe through geometry. An example is the general theory of relativity - an attempt to describe gravity based on geometry, which had stunning success in the last century. Going even further is string theory, in which geometry takes center stage in the form of six-dimensional Calabi-Yau figures. The book examines the ideas from geometry and physics needed to understand how Calabi-Yau manifolds came to be and why many physicists and mathematicians attach such importance to them. We tried to consider these varieties with different sides– their functional features; the calculations that led to their discovery; the reasons why string theorists find them attractive; and also the question of whether these figures are the key to understanding our Universe (and perhaps other universes too).
This is roughly how you can describe the purpose of this book. We can discuss the extent to which we succeeded in realizing our plans. But, without a doubt, nothing would have happened without the technical, editorial and emotional support of many people. There were too many to list them all, but I'll try to do so.
I have received immeasurable help from the persons already mentioned by my co-author. They are Eugenio Calabi, Simon Donaldson, Brian Greene, Tristan Hubsch, Andrew Strominger, Cumrun Vafa, Edward Witten, and especially Robert Greene, Bong Lian and Li-Sheng Tseng. It was the last three who provided me with mathematical advice as the book was being written, combining the art of clear explanation with amazing patience. In particular, it was Robert Greene, despite his busy schedule, who met with me twice a week to explain the features of differential geometry. Without his help, I would have found myself in extremely difficult situations countless times. Lian helped me get my head around geometry, and Tseng provided invaluable final touches to our ever-evolving manuscript.
Physicists Allan Adams, Chris Beasley, Shamit Kachru, Liam McAllister and Burt Ovrut answered my questions day and night, avoiding many failures. I cannot help but mention others who generously shared their time with me. These are Paul Aspinwall, Melanie Becker, Lydia Bieri, Volker Braun, David Cox, Frederik Denef, Robert Dijkgraaf, Ron Donagi, Mike Douglas, Steve Giddings, Mark Gross, Arthur Hebecker, Petr Horava, Matt Kleban, Igor Igor Klebanov, Albion Lawrence, Andrei Linde, Juan Maldacena, Dave Morrison, Lubos Motl, Hirosi Ooguri, Tony Pantev (Tony Pantev), Ronen Plesser, Joe Polchinski, Gary Shui, Aaron Simons, Raman Sundrum, Wati Taylor, Bret Underwood ( Bret Underwood), Deane Yang and Xi Yin.
This is just the very tip of the iceberg. I also had help from Eric Adelberger, Salem Ali, Bruce Allen, Nima Arkani-Hamed, Michael Atiyah, John Baez, Thomas Thomas Banchoff, Katrin Becker, George Bergman, Vincent Bouchard, Philip Candelas, John Coates, Andrea Cross, Lance Dixon Lance Dixon, David Durlach, Dirk Ferus, Felix Finster, Dan Freed, Ben Freivogel, Andrew Frey, Andreas Gutman Andreas Gathmann, Doron Gepner, Robert Geroch, Susan Gilbert, Cameron Gordon, Michael Green, Arthur Greenspoon, Marcus Grisaru Grisaru), Dick Gross, Monica Guica, Sergei Gukov, Alan Guth, Robert S. Harris, Matt Headrick, Jonathan Jonathan Heckman, Dan Hooper, Gary Horowitz, Stanislaw Janeczko, Lizhen Ji, Sheldon Katz, Steve Kleiman, Max Kreuser (Max Kreuzer), Peter Kronheimer, Mary Levin, Erwin Lutwak, Joe Lykken, Barry Mazur, William McCallum, John McGreevy ( John McGreevy, Stephen Miller, Cliff Moore, Steve Nahn, Gail Oskin, Rahul Pandharipande, Joaquin Perez, Roger Penrose Penrose), Miles Reid, Nicolai Reshetikhin, Kirill Saraikin, Karen Schaffner, Michael Schulz, John Schwarz, Ashoke Sen ), Kris Snibbe, Paul Shellard, Eva Silverstein, Joel Smoller, Steve Strogatz, Leonard Susskind, Yan Soibelman , Erik Swanson, Max Tegmark, Ravi Vakil, Fernando Rodriguez Villegas, Dwight Vincent, Dan Waldram, Devin Walker ), Brian Wecht, Toby Wiseman, Jeff Wu, Chen Ning Yang, Donald Zeyl and others.
It is difficult to illustrate many of the concepts in this book, but fortunately this problem was solved with the help of Xiaotian (Tim) Yin and Xianfeng (David) Gu from the Department of Computer Science at the University of Stony Brook, who in turn were assisted by Huayong Li and Wei Zeng. Also providing illustration assistance were Andrew Hanson (the primary visualizer of the Calabi-Yau manifold), John Orgea, and Richard Palais.
I would also like to thank my friends and family, including Will Blanchard, John DeLancey, Ross Eatman, Evan Hadingham, Harris McCarter and John Tibbetts (John Tibbetts), who read drafts of the book and provided advice and support. For their invaluable help in solving organizational issues, my co-author and I would like to thank Maureen Armstrong, Lily Chan, Hao Hu and Gena Bursan.
The text of this book contains references to materials from other publications. These are, in particular, “The Elegant Universe” by Brian Greene, “Euclid’s Window” by Leonard Mlodvinov and Robert Osserman’s books “Poetry of the Universe” and “The Cosmic Landscape” by Leonard Suskind, which have not yet been translated into Russian.
Our book would never have seen its readership if not for the help of John Brockman, Katinka Matson, Michael Healey, Max Brockman, Russell Weinberger and other collaborators Brockman Literary Agency, Inc. T. J. Kelleher of Basic Books believed in us and our book, and with the help of his colleague Whitney Casser, the publication became respectable. Kay Mariea, managing editor of Basic Books, oversaw all stages of the book's publication, and Patricia Boyd carried out literary editing. It was from her that I learned that “the same” is no different from “exactly the same”.
Finally, I would like to especially thank my family members Melissa, Juliet and Paulina, as well as my parents Lorraine and Marty, my brother Fred and my sister Sue. They all behaved as if six-dimensional Calabi-Yau manifolds were the most amazing thing that exists in our world, and did not even suspect that these manifolds were beyond its boundaries.
Steve Nadis, Cambridge, MA, March 2010 Introduction Shapes of Things to Come
God is a geometer.
Plato Around 360 BC, Plato wrote Timaeus, a history of creation told in the form of a dialogue between his teacher Socrates and three other participants: Timaeus, Critias, Hermocrates. Timaeus is a fictional character who came to Athens from the southern Italian city of Locri, “an expert in astronomy who made it his main business in order to understand the nature of the Universe.” In the mouth of Timaeus, Plato puts his own theory, in which geometry plays a central role.
Plato was fascinated by a group of convex figures, a special class of polyhedra called Platonic solids. The faces of each such body consist of identical regular polygons. For example, a tetrahedron has four regular triangular faces. A hexahedron, or cube, is made up of six squares. The octahedron consists of eight equilateral triangles, the dodecahedron consists of twelve pentagons, and the icosahedron consists of twelve triangles.
The three-dimensional figures called Platonic solids were not invented by Plato. To be honest, the name of their inventor is unknown. It is generally accepted that Plato's contemporary Theaetetus of Athens proved the existence of five and only five regular polyhedra. Euclid, in his Elements, gave a complete mathematical description of these forms.
Rice. 0.1. The five Platonic solids are: tetrahedron, hexahedron (or cube), octahedron, dodecahedron and icosahedron. The prefix indicates the number of faces - four, six, eight, twelve and twenty, respectively. What distinguishes them from all other polyhedra is the congruence of all faces, edges and angles (between two edges)
Platonic solids have several interesting properties, some of them are equivalent to ways of describing them. In each such polyhedron, the same number of edges converge at one vertex. And around the polyhedron you can describe a sphere that each vertex will touch - in the general case, this behavior is not typical for polyhedra. Moreover, the angles at which the edges meet at each vertex are always the same. The sum of the number of vertices and the number of faces is equal to the number of edges plus two.
Plato attached metaphysical significance to these bodies, which is why his name was associated with them. Moreover, convex regular polyhedra, as described in the Timaeus, are the essence of cosmology. In Plato's philosophy there are four main elements: earth, air, fire and water. If we could examine each of these elements in detail, we would notice that they are composed of miniature versions of the Platonic solids. The earth would thus consist of tiny cubes, air of octahedra, fire of tetrahedrons, and water of icosahedrons. “There remains one more, fifth construction,” wrote Plato in the Timaeus, referring to the dodecahedron. “God determined it for the Universe, using it as a model.”
From today's point of view, based on more than two millennia of scientific development, Plato's hypothesis looks doubtful. At present, no agreement has yet been reached as to what the Universe consists of - leptons and quarks, or hypothetical elementary particles of preons, or even more hypothetical strings. However, we know that it is not just earth, air, water and fire on the surface of a giant dodecahedron. We also stopped believing that the properties of elements are strictly described by the forms of Platonic solids.
On the other hand, Plato never claimed that his hypothesis was definitely true. He considered the Timaeus to be a “plausible account,” the best that could be offered at the time. It was assumed that descendants could improve the picture and even radically transform it. As Timaeus states in his reasoning, “... we should rejoice if our reasoning turns out to be no less plausible than any other, and, moreover, remember that both I, the reasoner, and you, my judges, are only people, and therefore we have to be content in such matters a plausible myth, without requiring more.”
Of course, Plato misunderstood many things, but if we consider his theses in more detail in a general sense, we will find that there is truth in them too. An eminent philosopher demonstrates perhaps the greatest wisdom in understanding that his hypothesis may turn out to be incorrect, but at the same time become the basis for another, correct theory. His polyhedra, for example, are remarkably symmetrical objects: the icosahedron and dodecahedron can be rotated in sixty ways (and this number, not coincidentally, represents twice the number of edges of each solid) without their appearance changing. By building a cosmology on these forms, Plato correctly assumed that symmetry must lie at the heart of any plausible description of nature. And if there ever is a real theory of the universe—one in which all the forces are unified and all the components obey a few rules—we will need to uncover the underlying symmetry, the simplifying principle on which everything else is built.
It hardly needs mentioning that the symmetry of solids is a direct consequence of their exact shape, or geometry. And it was here that Plato made his second major contribution: he not only realized that mathematics was the key to understanding the universe, but also demonstrated an approach called the geometrization of physics, a breakthrough made by Einstein. In a burst of foresight, Plato suggested that the elements of nature, their qualities and the forces acting between them could be the result of the influence of a colossal geometric structure hidden from us. The world we see may well be just a reflection of the underlying geometry, inaccessible to our perception. This knowledge is extremely dear to me, since it is closely related to mathematical proof, which brought me fame. This may seem far-fetched, but there is another way of geometric representation that relates to the above idea. However, you will see this as you read the book.
Chapter One The Universe is Out There The invention of the telescope and its subsequent improvement over the years helped confirm a fact that has now become an elementary truth: there is much in the Universe that is inaccessible to our observations. Indeed, according to the data available today, almost three-quarters of the material world exists in a mysterious, invisible form called dark energy. Most of the remainder, with the exception of only four percent that is ordinary matter (including you and me), is called dark matter. True to its name, this matter can be considered "dark" in every sense: it is difficult to see and equally difficult to understand.
The observable region of outer space is a ball with a radius of about 13.7 billion light years. This region is often called the Hubble volume, which, of course, does not imply that the Universe is limited by its boundaries. According to modern scientific data, the Universe is infinite, so that a straight line drawn from the point where we are in any given direction will extend to infinity.
True, there is a possibility that space is curved so much that the Universe is still finite. But even if this is so, this curvature is so small that, according to some theories, the Hubble volume accessible to our observation is no more than one of thousands of similar regions existing in the Universe.
And the Planck space telescope, recently launched into orbit, may show in the coming years that space consists of at least a million Hubble volumes, only one of which will ever be accessible to us. In general, I agree with astrophysicists, although I understand that some of the above numbers may be controversial. What is certain is that we are only seeing the tip of the iceberg.
On the other hand, microscopes, particle accelerators and various devices designed to obtain data about the microcosm continue to open up a “miniature” Universe, illuminating a previously inaccessible world of cells, molecules, atoms and even smaller objects. However, now these studies have ceased to surprise anyone. Moreover, we can expect that our telescopes will penetrate even deeper into outer space, and microscopes and other instruments will bring to light even more objects of the invisible world.
However, over the past decades, thanks to a number of advances in theoretical physics, as well as some advances in geometry that I was lucky enough to be involved in, we have been able to realize something even more amazing: the Universe is not only larger than we can see, but it may also contain more. (or even much greater) number of dimensions than the three spatial dimensions with which we are accustomed to dealing.
The statement I have made is difficult to take for granted, because if there is something that we can say with confidence about the world around us, something that our sensations tell us, starting from the first conscious moment and the first tactile experiences, it is this number of measurements. And that number is three. Not “three plus or minus one,” but precisely three. At least that's what it seemed like for a very long time. But still, it is possible (just possible) that in addition to these three, there are some additional dimensions, so small that we simply have not paid attention to them until now. And, despite their small size, they can play such an important role, the significance of which we can hardly appreciate, being in our familiar three-dimensional world.
It may not be easy to accept, but the past century has taught us that whenever we go beyond everyday experience, our intuition begins to fail us. Special theory relativity states that if we move fast enough, then time will begin to flow slower for us, and this in no way correlates with our everyday sensations. If we take an extremely small object, then, according to the requirements of quantum mechanics, we will not be able to tell exactly where it is. For example, if we want to experimentally determine whether an object is behind door A or behind door B, we will find that it is neither here nor there - in the sense that it, in principle, has no absolute location. (It's also possible for an object to be in both places at the same time!) In other words, many strange phenomena in our world are not only possible, but also quite real, and tiny hidden dimensions may be just one such reality.
If this idea is correct, then there must be something like a hidden universe, which is an essential fragment objective reality, located beyond our senses. This would be a real scientific revolution for two reasons. Firstly, the existence of extra dimensions - the main theme of science fiction for more than a hundred years - is in itself so amazing that it deserves to take pride of place among the greatest discoveries in the history of physics. And secondly, such a discovery would not be the completion of a physical theory, but, on the contrary, a starting point for new research. For just as a general gains a clearer picture of the battle by observing the progress of the battle from some elevated place, thereby taking advantage of the additional vertical dimension, so the laws of physics could acquire a clearer visual view and therefore become easier to understand when viewed from a higher dimensional perspective.
We are accustomed to moving in three main directions: north-south, west-east, up-down. (Or, if it is more convenient for the reader: right-left, forward-backward, up-down.) Wherever we walk and drive - be it a trip to the grocery store or a flight to Tahiti - our movement is always a superposition of movements in these three independent directions. The existence of exactly three dimensions is so familiar that even an attempt to imagine some additional dimension and understand where it can be directed seems futile. For a long time, it seemed that what we see is what we have. In fact, this is exactly what Aristotle argued more than two thousand years ago in his treatise “On the Heavens”: “A quantity divisible in one dimension is a line, in two a plane, in three a body, and besides these there is no other quantity, so how three dimensions are all dimensions.” In 150 AD, the astronomer and mathematician Claudius Ptolemy tried to strictly prove that the existence of four dimensions is impossible, arguing that it is impossible to construct four mutually perpendicular lines. The fourth perpendicular, according to his statement, would have to be “completely immeasurable and indefinable.” His argument, however, was not so much a rigorous proof as it was a reflection of our inability to imagine and depict anything in four dimensions.
For mathematicians, each dimension is a “degree of freedom” - an independent direction of movement in space. A fly flying above our heads is capable of moving in any direction allowed in the sky. If there are no obstacles in its path, then it has three degrees of freedom. Let us now imagine that a fly somewhere in a car park is stuck in fresh tar. While she is temporarily deprived of the ability to move, the number of her degrees of freedom is zero, and she is completely limited in her movements to one point - a world with zero dimension. But this creature is stubborn, and not without a struggle it still gets out of the tar, although it damages the wing in the process. Deprived of the ability to fly, the fly now has only two degrees of freedom and can only crawl around the parking lot. Sensing the approach of a predator - for example, a hungry frog - the heroine of our story seeks refuge in a rusty exhaust pipe. Now the fly has only one degree of freedom, at least during the time its movement is limited to the one-dimensional (linear) world of a narrow pipe.
But have we considered all the moving options? A fly can fly in the air, stick to tar, crawl along asphalt or move inside a pipe - can you imagine anything else? Aristotle or Ptolemy would say no, which may be true from the point of view of a not particularly enterprising fly, but for modern mathematicians, who find no convincing reason to stop at three dimensions, the matter does not stop there. On the contrary, they believe that to properly understand geometric concepts such as curvature or distance, they must be considered in all possible dimensions from zero to n, with n being a very large number. The coverage of the concept under consideration will be incomplete if we stop at three dimensions - the point is that if any rule or law of nature operates in space of any dimension, then such rules and laws are stronger and, most likely, more fundamental than statements that are valid only in special cases.
Even if the problem you're struggling with is only in two or three dimensions, looking at the problem in other dimensions may be the key to the solution. Let's return to our example of a fly flying in three-dimensional space and having three possible directions of movement, or three degrees of freedom. Now let's imagine another fly that moves freely in the same space; for her, like for the first fly, there are also exactly three degrees of freedom, but the system as a whole no longer has three, but six dimensions - six independent directions for movement.
It was very pleasant for me to reap the fruits of my labors and watch how others, following me, paved the way to those places that were inaccessible to me. And yet, despite all the success, something still haunted me. Deep down I was sure that this work must have not only mathematical, but also physical applications, although I could not say exactly which ones. Some of my confidence stemmed from the fact that the differential equations involved in Calabi's hypothesis - in the case of zero Ricci curvature - were Einstein's equations for empty space, corresponding to a Universe without additional vacuum energy, for which the cosmological constant would be zero. Currently, the cosmological constant is generally considered to be positive and associated with dark energy, which causes the Universe to expand. Moreover, Calabi-Yau manifolds were solutions to Einstein's differential equations, just as, for example, the unit circle is a solution to the equation x 2 +y 2 =0.
Of course, there are many more equations needed to describe Calabi-Yau spaces than to describe a circle, and the complexity of these equations is much higher, but the basic idea remains the same. Calabi-Yau manifolds not only satisfy Einstein's equations, they satisfy them in an extremely elegant way, which I, in particular, find amazing. All this gave me reason to hope for their applicability in physics. I just didn't know where exactly.
I had no choice but to try to explain to my friends and postdoc physicists the reasons why I believe the Calabi hypothesis and the so-called Yau's theorem so important for quantum gravity. The main problem was that at that time my understanding of the theory of quantum gravity was clearly insufficient for me to rely entirely on my own intuition. I returned to the idea from time to time, but mostly sat back and waited to see what would come of it.
As the years passed, while I and other mathematicians continued to work on the Calabi conjecture, trying to realize extensive plans for its application in the field of geometric analysis, there was also some behind-the-scenes movement in the world of physics that I was not aware of. This process began in 1984, which turned out to be a turning point for string theory, which that year began its rapid ascent from a speculative idea to a full-fledged theory.
Before describing these exciting developments, it is worth talking more about string theory itself, which boldly attempted to bridge the gap between general relativity and quantum mechanics. It is based on the assumption that the smallest particles of matter and energy are not point particles, but tiny, vibrating sections of strings, either closed in loops or open. Just like the strings of a guitar are capable of producing different notes, these fundamental strings are also capable of vibrating in a myriad of ways. String theory suggests that strings that vibrate differently correspond to different particles and forces found in nature. If this theory is correct, then the problem of unification of forces is solved as follows: all forces and particles are interconnected, since they are all manifestations of excitations of the same main string. You could say that this is exactly what the Universe is made of: when you go down to the most elementary level of the universe, you will find that everything is made of strings.
String theory borrows from the Kaluza-Klein theory the general idea that achieving a grand synthesis physical strength requires additional measurements. The proof is partly based on the same postulates: all four interactions existing in nature - gravitational, electromagnetic, weak and strong - simply do not have enough space in the four-dimensional theory. If we take Kaluza and Klein's approach and ask how many dimensions are needed to combine all four forces into a single theory, then given the five dimensions needed for gravity and electromagnetism, a couple of dimensions for the weak force, and a few more for the strong force, it turns out that the minimum number of dimensions is eleven. However, this is not entirely true - which, among other things, was shown by physicist Edward Witten.
Fortunately, string theory is not based on such an arbitrary treatment of physical concepts, such as choosing a random number of dimensions and expanding a matrix or Riemann metric tensor proportionally to it, followed by estimating how many and what forces will fit into this tensor. On the contrary, the theory precisely predicts the number of dimensions needed, and this number is ten - the four "ordinary" space-time dimensions examined by telescopes, plus six additional ones.
The reason why string theory requires exactly ten dimensions is quite complex and is based on the need to preserve symmetry - the most important condition construction of any fundamental theory - as well as the need to achieve compatibility with quantum mechanics, which is undoubtedly one of the key ingredients of any modern theory. But in essence the explanation boils down to the following: the greater the number of dimensions of a system, the greater the number of possible oscillations in it. To reproduce the full range of possibilities for our Universe, the number of permissible types of vibrations, according to string theory, must not only be very large, but also clearly defined - and this number can only be obtained in ten-dimensional space. We'll discuss another version or "generalization" of string theory later, called M-theory, which requires eleven dimensions, but we won't touch on that for now.
A string whose vibrations are limited to one dimension can only vibrate in longitudinal direction - by compression and stretching. In the case of two dimensions, vibrations of the string will arise as in longitudinal, and perpendicular to it transverse direction. For three or more dimensions, the number of independent oscillations will continue to grow until the dimension becomes ten (nine spatial dimensions and one time dimension) - exactly the case in which the mathematical requirements of string theory are satisfied. This is why string theory requires at least ten dimensions. Strictly speaking, the reason why string theory requires exactly ten dimensions, and no more and no less, has to do with the concept of reduction of anomalies, which takes us back to 1984, to the point where I left off.
Most string theories developed up to that point suffered from anomalies or incompatibilities that rendered their predictions meaningless. These theories, for example, led to the emergence of the wrong type of left-right symmetry - incompatible with quantum theory. The key breakthrough was made by Michael Green, then at Queen Mary's College in London, and John Schwartz of the California Institute of Technology. The main problem that Green and Schwartz managed to overcome related to the so-called parity violation- the idea that the fundamental laws of nature are asymmetrical with respect to specular reflection. Green and Schwartz discovered a way to formulate string theory in a way that implied that parity violation actually occurred in the system. The quantum effects that caused all sorts of inconsistencies in string theory were miraculously canceled out in ten-dimensional space, thereby raising hopes that this theory was the true one. The success of Green and Schwartz marked the beginning of what would later be called the first string revolution. The fact that they managed to avoid anomalies allowed us to talk about the ability of this theory to lead to an explanation of very real physical effects.
Shintan YAU, Steve NADIS
(Shing-TungYau, SteveNadis. The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions)
Legendary mathematician Shintan Yau argues that geometry is not only the basis of string theory, but also lies in the very nature of our Universe.
Eleventh chapter. Blooming Universe
(Everything you wanted to know about the end of the world, but were afraid to ask)
A man comes to the laboratory, where he is met by two physicists: a woman - a senior researcher and her assistant - a young man who shows the guest many research instruments that occupy the entire room: a stainless steel vacuum chamber, sealed containers with a refrigerant - nitrogen or helium, a computer , various measuring instruments, oscilloscopes, etc. A person is handed a control panel and told that the fate of the experiment, and perhaps the fate of the entire Universe, is now in his hands. If the young scientist does everything correctly, the device will receive energy from the quantized vacuum, giving humanity an unusually generous gift - the so-called “energy of creation in our hands.” But if the young scientist makes a mistake, his experienced colleague warns, the device could trigger a phase transition, causing the vacuum of empty space to decay to a lower energy state, releasing all the energy at once. A female physicist says that “this will not only be the end of the Earth, but the end of the entire Universe.” The man grips the control panel nervously, his palms sweaty. There are only a few seconds left until the moment of truth arrives. “You better decide quickly,” they tell him.
Although this is science fiction - an excerpt from the story "Vacuum States" by Jeffrey Landis - the possibility of vacuum decay is not complete fantasy. This question has been studied for a number of decades, as can be seen from publications in more serious scientific journals than Asimov's Science Fiction, namely in Nature, Physical Review Letters, Nuclear Physics B, etc., by scientists such as Sidney Coleman, Martin Rees , Michael Turner and Frank Wilczek. Currently, many physicists, and probably most people interested in similar questions, believe that the vacuum state of our Universe, that is, empty space devoid of any matter except for particles moving chaotically as a result of quantum fluctuations, is metastable rather than stable. If these theorists are right, then the vacuum will eventually collapse, which will have the most devastating consequences for the world (at least from our point of view), although these troubles may not be observed until the Sun disappears and evaporates black holes, protons will not decay.
Although no one knows what will happen in the long term, there seems to be one thing that many people agree on, at least in some scientific circles: the current structure of the world is not immutable, and eventually the collapse of the vacuum will occur. Refutations usually go like this: Although many researchers believe that a completely stable vacuum energy state or cosmological constant is not consistent with string theory, it should not be forgotten that string theory itself, unlike the mathematical statements that describe it, has not yet been proven. Moreover, I must remind readers that I am a mathematician, not a physicist, and we have touched on areas that are beyond my expertise. The question of what might ultimately happen to the six compact dimensions of string theory should be posed by physicists, not mathematicians. Since the death of these six dimensions may be associated with the death of part of our Universe, research of this kind necessarily involves an uncertain, even unreliable experiment, since, fortunately, we have not yet conducted a decisive experiment concerning the end of our Universe. And we don't have the resources, other than Landis's fertile imagination, to stage it.
With this in mind, if possible, approach this discussion with a healthy skepticism, using the approach I have chosen - as a fantastic leap in the land of probability. There will be a chance to find out what physicists think about what might happen to the six hidden dimensions that have been so much debated. We don't have any proof yet, and we don't even know how to test it, but I'll leave it to you to see how far imagination and competent speculation can take you.
Imagine that the scientist in Landis' story pressed a button on a remote control, suddenly setting off a chain of events that would lead to the collapse of the vacuum. What would happen then? But no one knows this. But regardless of the outcome - whether we have to go through fire or through ice (almost according to Robert Frost, who wrote: “Some say that the world will perish in fire, others in ice ...”) - our world, of course, must change beyond recognition. As Andrew Frey (McGill University) and colleagues wrote in an issue of Physical Review D in 2003: “one of the types of [vacuum] decay discussed in this paper would literally mean the end of the universe for anyone who had It’s unfortunate to witness this.” In this regard, there are two scenarios. Both involve radical changes to the status quo, although the first scenario is more severe because it entails the end of spacetime as we know it.
Let's recall the drawing from Chapter Ten, which shows a small ball rolling along a slightly curved surface in which the height of each point corresponds to different levels of vacuum energy. At the moment, our ball is in a semi-stable state, which is called a potential hole - by analogy with a small depression or hole in some hilly landscape. Let's assume that the bottom of this hole is above sea level, or, in other words, the vacuum energy value remains positive. If this landscape is classical, then the ball will remain in this hole indefinitely. In other words, his “resting place” will become his “final resting place.” But the landscape is not classic. This is the landscape of quantum mechanics, and interesting things can happen in this case: if the ball is extremely small and therefore obeys the laws of quantum mechanics, then it can literally drill through the side of the hole to reach the outside world - which is the result of a very real phenomenon known as quantum tunneling. It is possible thanks to fundamental uncertainty, one of the concepts of quantum mechanics. According to the uncertainty principle formulated by Werner Heisenberg, location, contrary to the real estate mantra, is not only a thing, and it is not even an absolute thing. And if there is the greatest probability of finding a particle in one place, then there is also a probability of finding it in other places. And if such a probability exists, the theory asserts, then, in the end, this event will occur if the wait is long enough. This principle is true for balls of all sizes, although a large ball is much less likely to be found elsewhere than a small ball.
Surprisingly, the effects of quantum tunneling can be observed in the real world. This well-studied phenomenon underlies the operation of scanning tunnel microscopes, when electrons pass through seemingly impenetrable barriers. For a similar reason, chip manufacturers cannot make chips too thin, otherwise the chips will be hampered by electron leakage due to tunneling effects.
The idea of particles, such as an electron, metaphorically or actually tunneling through a wall is one thing, but what about spacetime in general? The concept of vacuum tunneling during the transition from one energy state to another is admittedly difficult to understand, although the theory was well developed by Coleman and colleagues in the 1970s. In this case, the barrier is not a wall, but some kind of energy field that prevents the vacuum from moving to a state of lower energy, more stable, and therefore more preferable. The change in this case occurs due to phase transition similar to how liquid water turns into ice or steam, but it changes most of The universe, perhaps even that part of it where we live.
This brings us to the climax of the first scenario, in which the current vacuum state tunnels from a state with little positive energy (in fact, what is today called dark energy or the cosmological constant) into a state with negative energy. As a result, the energy that is currently causing our Universe to accelerate its expansion will compress it to a point, leading to a catastrophic event known as the Big Crunch. At such a cosmic singularity, both the energy density and the curvature of the Universe will become infinite, which is the same as if we suddenly fell into the center of a black hole or if the Universe returned to the state of the Big Bang.
The events that may follow the Big Crunch can be summed up in two words: "bet's off!" "We don't know what will happen to spacetime, let alone what will happen to extra dimensions," says physicist Steve Giddings of the University of California, Santa Barbara. It is beyond our experience and understanding in almost every way.
Quantum tunneling is not the only way to initiate a change in the vacuum state: this can be done using so-called thermal fluctuations. Let's go back to our tiny ball at the bottom of the potential hole. The higher the temperature, the faster atoms, molecules and other elementary particles move. And if the particles are moving, some of them may accidentally crash into the ball, pushing it in one direction or another. On average, these collisions balance out and the ball remains in a relatively stable position. But suppose that, in a statistically favorable situation, several atoms hit the ball sequentially, and in the same direction. As a result of the simultaneous action of several such collisions, the ball can be pushed out of the hole. He will roll down the inclined surface and will probably continue to roll until his energy becomes equal to zero, unless, of course, when moving, he does not end up in another hole or depression.
In honor of this, we are offering a 30% discount on this series, and below is an excerpt from the book “String Theory and the Hidden Dimensions of the Universe” by Shintan Yau and Steve Nadis - “Loops in Space-Time.”
Sigmund Freud believed that in order to understand the nature of the human mind, it is necessary to study people whose behavior does not fit into generally accepted norms, that is, is abnormal - people obsessed with strange, obsessive ideas: for example, his famous patients included “human -rat" (who had crazy fantasies in which people dear to him were tied by their buttocks to a pot of rats) and "wolfman" (who often dreamed of being eaten alive by white wolves sitting in a tree in front of his bedroom window). Freud believed that we learn most about typical behavior by studying the most unusual, pathological cases. Through such research, he said, we might eventually come to understand both norms and deviations from them.
We often take a similar approach in mathematics and physics. “We look for regions of space where classical descriptions do not work, because it is in these regions that we discover something new,” explains Harvard astrophysicist Avi Loeb. Whether we're talking about abstract space in geometry or the more material space we call the Universe, the regions "where something terrible happens to space, where things collapse," as Loeb says, are the regions we call singularities.
Contrary to what you might think about singularities, they are widespread in nature. They are all around us: a drop of water breaking off and falling from a faulty faucet is the most common example (often seen in my house), a place (well known to surfers) where ocean waves break and break, folds in a newspaper (which show is an article important or simply “water”) or the place of twists on a balloon rolled in the shape of a French poodle. “Without singularities, you cannot talk about forms,” notes geometer Heisuke Hironaka, professor emeritus Harvard University. He gives the example of his own signature: “If there are no intersecting lines or sharp corners, then these are just scribbles. The singularity would be lines intersecting or suddenly changing direction. There are many things like this in the world, and they make the world more interesting.”
In physics and cosmology, two types of singularities stand out among countless other possibilities. One kind is a singularity in time known as the Big Bang. As a geometrician, I don't know how to imagine the Big Bang because no one, including physicists, really knows what it is. Even Alan Guth, originator of the concept of cosmic inflation, a concept that he says "puts a bang within the Big Bang," admits that the term Big Bang has always suffered from vagueness, probably because "we still don't know (and "We may never know) what really happened." I believe that in this case modesty will not hurt us.
And while we are fairly ignorant when it comes to applying geometry to the exact moment of the birth of the universe, we geometers have made some progress in the fight against black holes. A black hole is essentially a piece of space compressed into a point by gravity. All this mass, packed into a tiny space, forms a super-dense object, the second cosmic velocity (a measure of its gravitational attraction) near which exceeds the speed of light, which leads to the capture of any matter, including light.
Even though the existence of black holes follows from Einstein's general theory of relativity, black holes still remain strange objects, and Einstein himself denied their existence until 1930, that is, 15 years after German physicist Karl Schwarzschild presented them in the form of solutions to Einstein's famous equations. Schwarzschild did not believe in the physical reality of black holes, but today the existence of such objects is a generally accepted fact. “Nowadays, black holes are discovered with amazing consistency every time someone at NASA needs another grant,” says Andrew Strominger.
And although astronomers have discovered a large number of black hole candidates and accumulated a wealth of observational data confirming this thesis, black holes are still shrouded in mystery.
General relativity provides a perfect and adequate description of large black holes, but the picture collapses when we move toward the center of the vortex and consider a vanishingly small singular point of infinite curvature.
General relativity can't deal with tiny black holes, smaller than a speck of dust, which is where quantum mechanics comes into play. The inadequacy of general relativity becomes glaringly obvious in the case of such miniature black holes, where the masses are enormous, the distances are tiny, and the curvature of spacetime cannot be imaged. In this case, string theory and Calabi-Yau space come to the rescue, which have been welcomed by physicists since the creation of the theory, in particular because they can resolve the conflict between adherents of general relativity and supporters of quantum mechanics.
One of the most heated debates between proponents of these distinguished branches of physics revolves around the question of the destruction of information by a black hole. In 1997, Stephen Hawking of the University of Cambridge and Kip Thorne of Caltech made a bet with John Preskill, also of Caltech. The subject of the dispute was the investigation theoretical discovery Hawking, made in the early 1970s, concluded that black holes are not completely “black”. Hawking showed that these objects have very low, but not zero, temperatures, meaning that they must retain some amount of thermal energy. Like any other “hot” body, a black hole will radiate energy into the external environment until all energy is completely exhausted and the black hole evaporates. If the radiation emitted by a black hole is strictly thermal and therefore devoid of information content, then the information initially retained within the black hole - say, if it absorbs a star with a certain composition, structure and history - will disappear when the black hole will evaporate. This conclusion violates the fundamental principle of quantum theory, which states that the information of a system is always conserved. Hawking argued that, contrary to quantum mechanics, information could be destroyed in the case of black holes, and Thorne agreed with him. Preskill argued that information would survive.
“We believe that if you throw two ice cubes into a pot of boiling water on Monday and test the water atoms on Tuesday, you will be able to determine that two ice cubes were thrown into the water the day before,” Strominger explains, “not practically, but basically". Another way to answer this question is to take a book like Fahrenheit 451 and throw it into the fire. “You may think that the information is lost, but if you have enough instruments and computer technology and you can measure all the parameters of the fire, analyze the ashes, and also resort to the services of “Maxwell’s demon” (or in this case “Laplace’s demon”), then you can reproduce the original state of the book,” notes physicist Hiroshi Oguri of Caltech.6 “However, if you threw the same book into a black hole,” Hawking counters, “the data would be lost.” Preskill, in turn, like Gerard 't Hooft and Leonard Suskind before him, defends the position that the two cases are not radically different from each other and that the radiation of a black hole must, in some subtle way, contain the information of Ray Bradbury's classic, which, theoretically, can be restored.
The stakes were high, since one of the cornerstones of science was at stake - the principle of scientific determinism. The idea of determinism is that if you have all the possible data describing a system at a particular time, and you know the laws of physics, then, in principle, you can determine what will happen to the system in the future, and also infer that what happened to her in the past. But if information can be lost or destroyed, then the principle of determinism loses its force. You cannot predict the future, you cannot draw conclusions about the past. In other words, if information is lost, then you are also lost. Thus, the stage was set for a decisive battle with the classics. “This was the moment of truth for string theory, which said it could reconcile quantum mechanics and gravity in an appropriate way,” says Strominger. “But could it explain the Hawking paradox?” Strominger discussed this issue with Cumrun Vafa in a groundbreaking article in 1996. To solve the problem, they used the concept of black hole entropy. Entropy is a measure of the randomness or disorder of a system, but also serves as a measure of the amount of information contained in the system. For example, imagine a bedroom with lots of shelves, drawers and counters, as well as various pieces of art displayed on the walls and hanging from the ceiling. Entropy refers to the number of different ways in which you can organize or disorganize all your things - furniture, clothes, books, pictures and various knickknacks in this room. To a certain extent, the number of possible ways to organize the same elements in a given space depends on the size of the room or its volume - the product of length, width and height. The entropy of most systems is related to their volume. However, in the early 1970s, physicist Jacob Bekenstein, then a graduate student at Princeton, proposed that the entropy of a black hole is proportional to the area of the event horizon surrounding the black hole, rather than to the volume contained within the horizon. The event horizon is often referred to as the point of no return, and any object crossing this invisible line in space will fall prey to the gravitational pull and inevitably fall into the black hole. But it's probably better to talk about the surface of no return, since in reality the horizon is a two-dimensional surface, not a point. For a non-rotating (or "Schwarzschild") black hole, the area of this surface depends solely on the mass of the black hole: the greater the mass, the larger area. The position that the entropy of a black hole—a reflection of all possible configurations of a given object—depends solely on the area of the event horizon implied that all configurations are located on the surface and that all information about the black hole is also stored on the surface. (We can draw a parallel with the bedroom in our previous example, where all the objects are located along the surfaces - walls, ceiling and floor, rather than floating in the center of the room in the internal space.)
Bekenstein's work, coupled with Hawking's ideas about black hole radiation, gave the world an equation for calculating the entropy of a black hole. Entropy, in accordance with the Bekenstein–Hawking formula, is proportional to the area of the event horizon. Or, more precisely, the entropy of a black hole is proportional to the area of the horizon divided by four Newtonian gravitational constants (G). This formula shows that the black hole, which is three times more massive than the Sun, has an astonishingly high entropy, on the order of 1078 joules per degree Kelvin. In other words, a black hole is extremely disordered.
The fact that a black hole has such staggeringly high entropy shocked scientists, given that in general relativity, a black hole is completely described by just three parameters: mass, charge and spin.
On the other hand, giant entropy implies enormous variability in the internal structure of a black hole, which must be specified by more than three parameters.
The question arises: where did this variability come from? What other things inside a black hole can change as much? The answer, apparently, lies in breaking the black hole into microscopic components, just as Austrian physicist Ludwig Boltzmann did with gases in the 1870s. Boltzmann showed that it was possible to infer the thermodynamic properties of gases from the properties of the constituent individual molecules. (There are actually a lot of these molecules, for example in one bottle ideal gas under normal conditions there are approximately 1022 molecules.) Boltzmann's idea turned out to be remarkable for many reasons, including the fact that he came to it decades before the existence of molecules was confirmed. Given the huge number of gas molecules, Boltzmann argued that the average speed of movement, or the average behavior of individual molecules, determines the general properties of the gas - volume, temperature and pressure, that is, the properties of the gas as a whole. Thus, Boltzmann formulated a more accurate idea of the system, stating that the gas is not a solid body, but consists of many particles. A new look at the system allowed him to give a new definition of entropy as the statistical weight of a state - the number of possible microstates (ways) by which one can go to a given macroscopic state. Mathematically, this position can be formulated as follows: entropy (S) is proportional to the natural logarithm of the statistical weight. Or, equivalently, the statistical weight is proportional to eS.
The approach that Boltzmann pioneered is called statistical mechanics, and about a century later people tried to interpret black holes using statistical mechanics methods. Twenty years after Bekenstein and Hawking posed this problem, it still has not been solved. All that was needed to solve it was "a microscopic theory of black holes, deducing the laws of black holes from some fundamental principles - analogous to Boltzmann's derivation of the thermodynamics of gases," Strominger says. Since the 19th century, it has been known that every system has an entropy associated with it, and from Boltzmann's definition of entropy it followed that the entropy of a system depends on the number of microstates of the system's components. “It would be a profound and distressing asymmetry if the relationship between entropy and the number of microstates were true for any system in nature except a black hole,” Strominger adds. Moreover, according to Oguri, these microstates are “quantized” because this is the only way one can hope to obtain a countable number of them. You can put a pencil on the table in an infinite number of ways, just as there are an infinite number of possible settings across the entire spectrum electromagnetic radiation. But as we mentioned in Chapter Seven, radio frequencies are quantized in the sense that radio stations transmit on a select number of discrete frequencies. The energy levels of a hydrogen atom are similarly quantized, so you can't choose arbitrary value; Only certain energy values are allowed. “Part of the reason Boltzmann had such a hard time convincing other scientists of his theory was that he was ahead of his time,” says Oguri. “Quantum mechanics was not developed until half a century later.”
This was the problem that Strominger and Vafa took on to solve. This was truly a test of string theory, since the problem involved the quantum states of black holes, which Strominger called "the quintessence of gravitational objects." He felt that it was his duty to solve this problem by calculating entropy, or admit that string theory was wrong.
The plan that Strominger and Vafa came up with was to calculate the entropy value using quantum microstates and compare it with the value calculated by the Bekenstein-Hawking formula, which was based on general relativity. Although the problem was not new, Strominger and Vafa used new tools to solve it, drawing not only on string theory, but also on Joseph Polchinski's discovery of D-branes and the emergence of M-theory - both events that took place in 1995, a year before the release their articles. “Polchinsky pointed out that D-branes carry the same type of charge as black holes and have the same mass and tension, so they look and smell the same,” notes Harvard physicist Hee Ying. “But if you can use one to calculate the properties of another, such as entropy, then there is more than a passing similarity.” This is the approach Strominger and Vafa took, using these D-branes to build new kinds of black holes, guided by string theory and M-theory.
The ability to construct black holes from D-branes and strings (the latter being a one-dimensional version of D-branes) results from the "dual" description of D-branes. In models where the efficiency of all forces acting on branes and strings (including gravity) is low (which is called weak connection), branes can be thought of as thin, membrane-like objects that have weak impact on the space-time around them and, therefore, bear little resemblance to black holes. On the other hand, with strong coupling and high interaction strength, branes can become dense, massive objects with an event horizon and a strong gravitational influence - in other words, objects indistinguishable from black holes.
However, it takes more than a heavy brane or many heavy branes to create a black hole. You also need some way to stabilize it, which is easiest to do, at least in theory, by wrapping the brane around something stable that doesn't shrink. The problem is that an object that has high tension (expressed as mass per unit length, area, or volume) can shrink to such a small size that it almost disappears, without having the appropriate structure to stop the process, much like an ultra-tight the rubber band shrinks into a tight ball when left to its own devices.
The key ingredient was supersymmetry, which, as discussed in Chapter Six, has the property of preventing the ground or vacuum state of a system from falling into ever lower energy levels. Supersymmetry in string theory often implies Calabi–Yau manifolds because such spaces automatically include this feature. So the challenge is to find stable subsurfaces within Calabi-Yau manifolds to wrap into branes. These subsurfaces, or submanifolds, which have less dimension than space itself, are sometimes called cycles (a concept introduced earlier in the book), which can sometimes be thought of as an incompressible loop around or through part of a Calabi-Yau manifold. In technical terms, a loop is a one-dimensional object, but loops involve more dimensions and can be thought of as higher-dimensional, incompressible "loops."
Physicists tend to think that the loop depends only on the topology of the object or hole that you can wrap around, regardless of the geometry of that object or hole. “If you change the shape, the cycle remains the same, but you get a different submanifold,” Yin explains. He adds that since this is a property of topology, the cycle itself cannot do anything to the black hole. “It’s only when you wrap one or more branes around a cycle that you can start talking about a black hole.” To ensure stability, the object you are wrapping with - be it a brane, string or rubber band - must be tight, without any folds. The loop you wrap around should be the smallest possible length or area. Laying a rubber band around a uniform, cylindrical pole is not an example of a stable situation because the band can easily be moved from side to side. At the same time, if the pole has different thicknesses, then stable cycles, which in this case are circles, can be found at the points of local minimum of the pole diameter, where the rubber band will not creep from side to side.
To make an analogy with Calabi-Yau manifolds, instead of a smooth pole, it is better to imagine another object that we wrap with a rubber band, such as a grooved pole or a donut of variable thickness, on which the minimum cycles will correspond to places where the diameter has a local minimum. There are different kinds of cycles around which a brane can be wrapped inside Calabi-Yau manifolds: these can be circles, spheres or tori of different dimensions, or Riemann surfaces of a high genus. Since branes carry mass and charge, the problem is to calculate the number of ways they can be placed into stable configurations within the Calabi-Yau manifold such that their resulting mass and charge are equal to the mass and charge of the black hole itself. “Even though these branes are wrapped separately, they still stick together to the interior of the [Calabi–Yau] and can be considered parts of a larger black hole,” explains Yin. There is an analogy that I admit is quite unappetizing, but I didn’t come up with it. I heard it from a Harvard physicist, whose name I will not name, and I am sure that he will also deny it, blaming the authorship on someone else. The situation in which individual wrapped branes stick together to form a larger object can be compared to a wet shower curtain with different strands of hair stuck to it. Each strand of hair is like an individual brane that is attached to a larger object, a shower curtain, similar to the brane itself. Even though each hair could be considered a separate black hole, they are all stuck together - stuck to the same sheet - making them part of one big black hole. Calculating the number of cycles, that is, calculating the number of ways to arrange D-branes, is a problem in differential geometry, since the number you get from this calculation corresponds to the number of solutions to the differential equation.
Strominger and Vafa transformed the problem of calculating the microstates of a black hole and, accordingly, calculating entropy into geometric problem: How many ways are there to place D-branes into Calabi-Yau manifolds to obtain the desired mass and charge? And this problem, in turn, can be expressed in terms of cycles: how many spheres and objects of other forms of the minimum size around which a brane can be wrapped can be placed inside a Calabi-Yau manifold? The answer to both of these questions obviously depends on the geometry of the given Calabi-Yau manifold. If you change the geometry, you change the number of possible configurations, or the number of spheres.
This is the big picture, and the calculation itself was still complex, so Strominger and Vafa spent a lot of time looking for a specific approach to this problem, that is, a way that would actually solve it.
They took on a very specific case and for their first attempt chose a five-dimensional interior space constructed by the direct product of a four-dimensional K3 surface and a circle. They also constructed a five-dimensional black hole located in flat five-dimensional space, to which they could compare a structure built from D-branes. This was no ordinary black hole. She had special properties, which were selected to make the problem "manageable": this black hole was both supersymmetric and extremal - the latter term meaning that it had the minimum possible mass for a given charge. We have already touched on supersymmetry, but it makes sense to talk about supersymmetry of a black hole only if the main vacuum in which it is located also preserves supersymmetry. This is not true in the low-energy region that we inhabit and where we cannot see supersymmetry in the particles around us. We cannot see it in the black holes that astronomers observe.
Once Strominger and Vafa modeled the black hole, they were able to use the Bekenstein–Hawking formula to calculate entropy based on the area of the event horizon. Next step was a calculation of the number of ways to configure D-branes in the interior so that this number corresponds to the design of a black hole of a given resultant charge and mass. Then the entropy calculated in this way, equal to the logarithm of the number of states, was compared with the entropy value obtained from the area of the event horizon, and the entropy values coincided. “They wiped everyone’s noses by getting a four in the denominator, and Newton’s constant, and everything else,” says Harvard physicist Frederic Denef. Denef adds that after twenty years of trying, “we finally have the first calculation of the entropy of a black hole using statistical mechanics methods.”
This was the main success of Strominger and Vafa, and also the success of string theory. Yin explained that the connection between D-branes and black holes has received a strong argument in its favor, and, in addition, two physicists have shown that the description of D-branes itself is fundamental. “You're probably wondering, can a brane be broken down into its components? Is it built from smaller particles? We are now confident that the brane does not have any additional structures, because physicists got the entropy right, and entropy, by definition, is proportional to the number of all states.”16 If the brane were composed of different particles, then it would have more degrees of freedom and, therefore, more combinations that would need to be taken into account when entropy calculation. But the result obtained in 1996 shows that this is not the case. Bran is all there is. Although branes of different numbers of dimensions look different, none of them have subcomponents and cannot be resolved into their components. Likewise, string theory holds that a string—the one-dimensional brane in M-theory—is all there is, and cannot be divided into smaller parts. Although the correspondence between two very different methods of calculating entropy was greeted with enthusiasm, it raised eyebrows. “At first glance, the black hole information paradox appears to have nothing to do with Calabi-Yau manifolds,” says physicist Aaron Simons of Brown University. “But the key to answering this question turned out to be the calculation of mathematical objects inside the Calabi-Yau manifold.”
Strominger and Vafa did not fully resolve the information paradox, although the detailed description of a black hole they arrived at through string theory showed exactly how information might be stored. Oguri said they had completed the most important first step of the study, "showing that the entropy of a black hole is the same as that of other macroscopic systems," including the burning book from our previous example. Both contain information that is, at least potentially, recoverable.
Of course, the 1996 results were just the beginning, since the first entropy calculation had little to do with real astrophysical black holes. The black holes in the Strominger–Vaff model, unlike those we see in nature, were supersymmetric—a condition necessary for the calculation to work. However, these results can be extended to non-supersymmetric black holes. As Simons explains: “Regardless of supersymmetry, all black holes contain a singularity. This is their main defining feature, and for this reason they are “paradoxical”. In the case of supersymmetric black holes, string theory has helped us understand what happens around this singularity, and the hope is that the result will not depend on whether the object is supersymmetric or not."
Additionally, a 1996 paper describes the artificial case of a compact five-dimensional interior space and a flat, non-compact five-dimensional exterior space. But spacetime is not usually considered in this way in string theory. The question is whether this model applies to the more common model: a six-dimensional internal space and a black hole located in a flat, four-dimensional space? The answer was given in 1997, when Strominger, along with Juan Maldacena - then a Harvard physicist, and Edward Witten - published a paper on their first work, which used the more familiar arrangement of six-dimensional internal space (Calabi-Yau, of course) and extended four-dimensional spacetime .
Reproducing the entropy calculation for a three-dimensional Calabi-Yau manifold, Maldacena said that "the spaces you put branes in have weaker supersymmetry" and are therefore closer to real world, and “the space in which you put black holes has four dimensions, which is consistent with our assumptions.” Moreover, the agreement with the Bekenstein–Hawking calculation was even stronger because, as Maldacena explains, calculating entropy from the area of the event horizon is only accurate when the event horizon is very large and the curvature is very small. As the size of black holes shrinks, and with it the surface area, the general relativity approximation becomes worse and it is necessary to introduce “corrections for quantum gravity"into Einstein's theory. While the original paper considered only "large" black holes - large compared to the Planck scale - for which it was sufficient to take into account the effects following from general relativity - the so-called first order term, the 1997 calculation also produced the first quantum term in addition to the first gravitational one. In other words, agreement between two different ways entropy calculation has become much better. In 2004, Oguri, Strominger, and Vafa went even further, generalizing the 1996 results to any kind of black hole that can be constructed by wrapping a brane around a cycle in a regular Calabi-Yau threefold, regardless of its size, and hence, regardless of the contribution of quantum mechanical effects. The authors of the article showed how to calculate quantum corrections to the theory of gravity not only for the first few terms, but also for the entire series containing an infinite number of terms.21 Vafa explained that by adding new terms to the expansion, “we got a more accurate method of calculation and more precise answer and, fortunately, even stronger agreement than before.”22 This is exactly the approach we usually try to take in mathematics and physics: if we find something that works under special conditions, we try to consider the more general case of whether it will work under less stringent conditions, and accordingly determine how far can we go.
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