ALSO BY AMIR D. ACZEL FERMAT'S LAST THEOREM Unlocking the Secret of an Ancient Mathematical Problem "Captivating...a tale of buried treasure...lucidly presented." —The New York Times Available from Dell PRAISE FOR AMIR D. ACZEL'S FERMAT'S LAST THEOREM "This is a captivating volume....The brilliant backdoor method used by Mr. Wiles as he reached his solution, along with the debt he owed to many other contemporary mathematicians, is graspable in Mr. Aczel's lucid prose. Equally important is the sense of awe that Mr. Aczel imparts for the hidden, mystical harmonies of numbers, and for that sense of awe alone, his slender volume is well worth the effort." —The New York Times "For more than three centuries, Fermat's Last Theorem was the most famous unsolved problem in mathematics; here's the story of how it was solved....An excellent short history of mathematics, viewed through the lens of one of its great problems—and achievements." —Kirkus Reviews "This exciting recreation of a landmark discovery reveals the great extent to which modern mathematics is a collaborative enterprise....While avoiding technical details, Aczel maps the strange, beautiful byways of modern mathematical thought in ways the layperson can grasp." —Publishers Weekly "A mathematical bonbon of a book...Briefly chronicles the history of the famous problem of the title, which was recently solved by a mathematician named Andrew Wiles after he had devoted seven years to the task....Aczel does a superb job of creating in the nonmathematical reader the illusion of comprehension." —The New Yorker Einstein, Relativity, and the Expanding Universe <> Amir D. Aczel A Delta Book Published by Dell Publishing a division of Random House, Inc. 1540 Broadway New York, New York 10036 Copyright © 1999 Amir D. Aczel. Cover design by Royce Becker. Cover photo © Illustrated London News/Archive Photos. All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without the written permission of the Publisher, except where permitted by law. For information address: Four Walls Eight Windows. Dell books may be purchased for business or promotional use or for special sales. For information please write to: Special Markets Department, Random House, Inc., 1540 Broadway, New York, N.Y. 10036. Delta® is a registered trademark of Random House, Inc., and the colophon is a trademark of Random House, Inc. Albert Einstein™ Licensed by the Hebrew University of Jerusalem. Represented by the Roger Richman Agency, Inc., Beverly Hills, CA. ISBN: 0-385-33485-0 Library of Congress Catalog Card Number: 99-36319 Reprinted by arrangement with Four Walls Eight Windows Manufactured in the United States of America Published simultaneously in Canada December 2000 10 9 8 7 6 5 4 3 2 FFG To my father, Captain E. L. Aczel Contents Preface < IX > CHAPTER 1 Exploding Stars < 1 > CHAPTER 2 Early Einstein < 1 3 > CHAPTER 3 Prague, 1911 < 2 7 > CHAPTER 4 Euclid's Riddle < 4 3 > CHAPTER 5 Grossmann's Notebooks < 6 1 > CHAPTER 6 The Crimean Expedition < 7 1 > CHAPTER 7 Riemann's Metric < 9 1 > CHAPTER 8 Berlin < 1 0 5 > CHAPTER 9 Principe Island < 1 2 1 > CHAPTER 10 The joint Meeting < 1 3 9 > CHAPTER 11 Cosmological Considerations < 1 4 9 > CHAPTER 12 The Expansion of Space < 1 6 7 > CHAPTER 13 The Nature of Matter < 1 8 1 > CHAPTER 14 The Geometry of the Universe < 1 8 9 > CHAPTER 15 Batavia, Illinois, May 4,1998 < 1 9 7 > CHAPTER 16 God’s Equation < 2 0 7 > References < 2 2 1 > Index < 2 2 5 > < VI I > Preface In January 1998, the way we perceive the universe changed forever. Astronomers found evidence that the cosmos is expanding at an ever-increasing rate. As soon as the new findings were announced, cosmologists from all over the world rushed to try to explain the underlying phenomenon. The most promising theory these scientists could come up with was one that Albert Einstein had proposed eight decades earlier and quickly retracted, calling it his greatest blunder. Every year, new developments prove the accuracy of Einstein's theories. But if the cosmologists' new assessments are correct, then Einstein was right even when he was sure he was wrong. About the time this astounding news was being reported, I received a curious piece of mail. It was sent to me by L. P. Lebel, a reader of my book Fermat's Last Theorem who had become a friend, and with whom I'd been exchanging letters. This time, however, the envelope contained no letter: simply a cutting of an article written by George Johnson in the New York Times. I read the article with great interest: it was about pure mathematics, not physics or cosmology. In the article, Mr. Johnson posed an intriguing question: Is it possible that other forms of mathe- < I X >