SCIENCE/MATHEMATICS SHING-TUNG $30.00 US / $36.00 CAN Praise for YAU & and the STEVE NADIS STRING THEORY THE SHAPE OF tring theory—meant to reconcile the INNER SPACE incompatibility of our two most successful GEOMETRY of the UNIVERSE’S theories of physics, general relativity and “The Shape of Inner Space provides a vibrant tour through the strange and wondrous possibility INNER SPACE THE quantum mechanics—holds that the particles that the three spatial dimensions we see may not be the only ones that exist. Told by one of the Sand forces of nature are the result of the vibrations of tiny masters of the subject, the book gives an in-depth account of one of the most exciting HIDDEN DIMENSIONS “strings,” and that we live in a universe of ten dimensions, and controversial developments in modern theoretical physics.” —BRIAN GREENE, Professor of © Susan Towne Gilbert © Susan Towne four of which we can experience, and six that are curled up Mathematics & Physics, Columbia University, SHAPE in elaborate, twisted shapes called Calabi-Yau manifolds. Shing-Tung Yau author of The Fabric of the Cosmos and The Elegant Universe has been a professor of mathematics at Harvard since These spaces are so minuscule we’ll probably never see 1987 and is the current department chair. Yau is the winner “Einstein’s vision of physical laws emerging from the shape of space has been expanded by the higher them directly; nevertheless, the geometry of this secret dimensions of string theory. This vision has transformed not only modern physics, but also modern of the Fields Medal, the National Medal of Science, the realm may hold the key to the most important physical mathematics. Shing-Tung Yau has been at the center of these developments. In this ambitious book, Crafoord Prize, the Veblen Prize, the Wolf Prize, and a phenomena we know about. written with experienced science writer Steve Nadis, Yau tells of his own mathematical experiences, Before there could be the physics of those Calabi-Yau MacArthur Fellowship. A member of the National Academy and of the parallel efforts in string theory and in mathematics to understand the nature of space.” of Sciences, Yau has written and edited more than twenty —JOE POLCHINSKI, Professor of Physics, manifolds, however, there had to be the math. In 1976, books. He lives in Cambridge, Massachusetts. University of California, Santa Barbara, author of String Theory, Vols. 1 & 2 Shing-Tung Yau, a geometer now based at Harvard, proved the mathematical existence of those intricate “The Shape of Inner Space has a distinctive style: in part autobiography, in part an account of shapes. It was work that helped earn Yau the Fields Medal, developments in geometric analysis and string theory over the past forty years, and comments mathematics’ highest honor, and, although no one knew it on future directions. It gives a unique insight into the thoughts of one of the most important and OF at the time, Yau had laid the ground on which a revolution influential mathematicians of our times.” —SIMON DONALDSON, Royal Society Research Professor in Pure Mathematics in science would eventually play out. and President of the Institute for Mathematical Science, Imperial College London The Shape of Inner Space traces the roots of Yau’s efforts STRING back to Einstein and Plato—both of whom considered “Shing-Tung Yau and Steve Nadis take the reader on a fascinating tour of many contemporary topics THEORY geometry crucial to any understanding of the physical 6.25 x 9.5 in geometry and physics. Readers will find many challenging ideas to explore in this book, and even Bulk: 1-1/16” and the world. Yau introduces his readers to the geometry specialists will enjoy Yau’s reminiscences about his education and work.” —EDWARD WITTEN, Spine: 1-5/16” Professor, Institute for Advanced Study GEOMETRY and topology needed to understand his work on the © Juliet Nadis of the Calabi-Yau manifold, and subsequent developments in the Basic Books Steve Nadis is a contributing “A fascinating first-hand account of how the math underlying string theory was discovered. Fields UNIVERSE’S field. And he explains the tremendous impact his proof Hardcover medalist Yau and ace science writer Nadis have teamed up to show the rest of us the deep geometry editor to Astronomy magazine. A graduate of Hampshire HIDDEN has had as it spread through the interconnected worlds that just might lie at the heart of the universe. It’ll twist you into knots of pleasure!” 4/color process College, he has written or contributed to more than two DIMENSIONS of math and physics, with insights from each field feeding —STEVEN STROGATZ, New York Times Opinionator dozen books. Nadis has also been a research fellow at MIT, contributor and professor of mathematics, Cornell University directly into the efforts of the other. In the end, he asks Finish: a staff researcher at the Union of Concerned Scientists, what this discovery means not only for making sense of our matte poly and a consultant to the World Resources Institute, the universe, but for contemplating its destiny and the future $30.00 US / $36.00 CAN Woods Hole Oceanographic Institute, and WGBH/NOVA. ISBN 978-0-465-02023-2 of geometry itself. 5 3 0 0 0 He lives in Cambridge, Massachusetts. SHING-TUNG YAU The Shape of Inner Space is the story of how physics met geometry, and the new picture of the universe that has Jacket design by Ariana Abud WINNER OF THE FIELDS MEDAL Jacket image © Andrew J. Hanson/Indiana University emerged as a result. It’s exciting; it’s challenging; and it’s A Member of the Perseus Books Group 9 7 8 0 4 6 5 0 2 0 2 3 2 and STEVE NADIS a story that’s never been told before. 08/10 www.basicbooks.com 0465020232-Yau_Layout 1 6/23/10 10:40 AM Page i THE SHAPE OF INNER SPACE 0465020232-Yau_Layout 1 6/23/10 10:40 AM Page ii 0465020232-Yau_Layout 1 6/23/10 10:40 AM Page iii The SHAPE of INNER SPACE String Theory and the Geometry of the Universe’s Hidden Dimensions Shing-Tung Yau and Steve Nadis Illustrations by Xianfeng (David) Gu and Xiaotian (Tim) Yin A MEMBER OF THE PERSEUS BOOKS GROUP New York 0465020232-Yau_Layout 1 6/23/10 10:40 AM Page iv Copyright © 2010 by Shing-Tung Yau and Steve Nadis Published by Basic Books, A Member of the Perseus Books Group All rights reserved. Printed in the United States of America. No part of this book may be reproduced in any manner whatsoever without written permission except in the case of brief quotations embodied in critical articles and reviews. For information, address Basic Books, 387 Park Avenue South, New York, NY 10016-8810. Books published by Basic Books are available at special discounts for bulk purchases in the United States by corporations, institutions, and other organizations. For more information, please contact the Special Markets Department at the Perseus Books Group, 2300 Chestnut Street, Suite 200, Philadelphia, PA 19103, or call (800) 810-4145, ext. 5000, or e-mail [email protected]. Designed by Timm Bryson Library of Congress Cataloging-in-Publication Data Yau, Shing-Tung, 1949- The shape of inner space : string theory and the geometry of the universe’s hidden dimensions / Shing-tung Yau and Steve Nadis ; illustrations by Xianfeng (David) Gu and Xiaotian (Tim) Yin. p. cm. Includes bibliographical references and index. ISBN 978-0-465-02023-2 (alk. paper) 1. Hyperspace. 2. String models. 3. Fourth dimension. I. Nadis, Steven J. II. Title. QA691.Y38 2010 530.1—dc22 2010009956 10 9 8 7 6 5 4 3 2 1 0465020232-Yau_Layout 1 6/23/10 10:40 AM Page v CONTENTS “Space/Time” (Poem) vii Preface ix Prelude: The Shapes of Things to Come xvii 1 A Universe in the Margins 1 2 Geometry in the Natural Order 17 3 A New Kind of Hammer 39 4 Too Good to Be True 77 5 Proving Calabi 103 6 The DNA of String Theory 121 7 Through the Looking Glass 151 8 Kinks in Spacetime 183 9 Back to the Real World 199 10 Beyond Calabi-Yau 227 11 The Universe Unravels 253 v 0465020232-Yau_Layout 1 6/23/10 10:40 AM Page vi vi Contents 12 The Search for Extra Dimensions 269 13 Truth, Beauty, and Mathematics 289 14 The End of Geometry? 307 Epilogue Another Day, Another Donut 321 Postlude Entering the Sanctum 325 “A Flash in the Middle of a Long Night” (Poem) 329 Notes 331 Glossary 345 Index 359 0465020232-Yau_Layout 1 6/23/10 10:40 AM Page vii SPACE/TIME Time, time why does it vanish? All manner of things what infinite variety. Three thousand rivers all from one source. Time, space mind, matter, reciprocal. Time, time it never returns. Space, space how much can it hold? In constant motion always in flux. Black holes lurking mysteries afoot. Space and time one without bounds. Infinite, infinite the secrets of the universe. Inexhaustible, lovely in every detail. Measure time, measure space no one can do it. Watched through a straw what’s to be learned has no end. Shing-Tung Yau Beijing, 2002 0465020232-Yau_Layout 1 6/23/10 10:40 AM Page viii 0465020232-Yau_Layout 1 6/23/10 10:40 AM Page ix PREFACE Mathematics is often called the language of science, or at least the language of the physical sciences, and that is certainly true: Our physical laws can only be stated precisely in terms of mathematical equations rather than through the written or spoken word.