Time Travel and Warp Drives Time Travel and Warp Drives A Scientifi c Guide to Shortcuts through Time and Space Allen Everett and Thomas Roman The University of Chicago Press Chicago and London allen everett is professor emeritus of physics at Tufts University. tom roman is a professor in the Mathematical Sciences Department at Central Connecticut State University. Both have taught undergraduate courses in time-travel physics. The University of Chicago Press, Chicago 60637 The University of Chicago Press, Ltd., London © 2012 by The University of Chicago All rights reserved. Published 2012. Printed in the United States of America 21 20 19 18 17 16 15 14 13 12 1 2 3 4 5 isbn-13: 978-0-226-22498-5 (cloth) isbn-10: 0-226-22498-8 (cloth) Library of Congress cataloging-in-Publication Data Everett, Allen. Time travel and warp drives : a scientifi c guide to shortcuts through time and space / Allen Everett and Thomas Roman. p. cm. Includes bibliographical references and index. isbn-13: 978-0-226-22498-5 (cloth : alk. paper) isbn-10: 0-226-22498-8 (cloth : alk. paper) 1. Time travel. 2. Space and time. I. Roman, Thomas. II. Title. qc173.59.s65e94 2012 530.11—dc23 2011025250 This paper meets the requirements of ansi/niso z39.48–1992 (Permanence of Paper). To my loving wife, Cecilia, and to my parents ( T. R.) In memory of my late beloved wife and cherished best friend, Marylee Sticklin Everett. For more than 42 years of love, companionship, support, and wonderful memories, thank you. ( A. E.) Contents Preface > ix Acknowledgments > xi 1 Introduction > 1 2 Time, Clocks, and Reference Frames > 10 3 Lorentz Transformations and Special Relativity > 22 4 The Light Cone > 42 5 Forward Time Travel and the Twin “Paradox” > 49 6 “Forward, into the Past” > 62 7 The Arrow of Time > 76 8 General Relativity: Curved Space and Warped Time > 89 9 Wormholes and Warp Bubbles: Beating the Light Barrier and Possible Time Machines > 112 10 Banana Peels and Parallel Worlds > 136 11 “Don’t Be So Negative”: Exotic Matter > 158 12 “To Boldly Go . .”? > 181 13 Cylinders and Strings > 196 14 Epilogue > 218 Appendix 1. Derivation of the Galilean Velocity Transformations > 225 Appendix 2. Derivation of the Lorentz Transformations > 227 Appendix 3. Proof of the Invariance of the Spacetime Interval > 232 Appendix 4. Argument to Show the Orientation of the x',t' Axes Relative to the x,t Axes > 234 Appendix 5. Time Dilation via Light Clocks > 236 Appendix 6. Hawking’s Theorem > 241 Appendix 7. Light Pipe in the Mallett Time Machine > 250 Bibliography > 253 Index > 259 Preface In part, our motivation for writing this book is the classes that we have taught on the subject at our respective universities, Tufts (A. E.) and Central Connecticut State (T. R.). Many, but not all, of our students were science fi ction buff s. They ranged from present or prospective physics majors to fi ne arts majors; several of the latter did very well and were among the most fun to teach. The courses aff orded us an opportunity, unusual for theoretical physicists, to give undergraduates some access to our own re- search, using essentially no mathematics beyond high school algebra. We are grateful to all of the students in those classes over the years for their enthusi- asm and intellectual stimulation. Our aim here was to write a book for people with diff erent levels of math and physics backgrounds, skills, and interests. Since we believe that what cur- rently is on off er is either too watered down or too sensationalistic, we decided to try our hand. The level of this book is intended for a person who is perhaps a Star Trek fan or who likes to read Scientifi c American occasionally, but who fi nds it not detailed enough for a good understanding of the subject matter. We as- sume that our reader knows high school algebra, but no knowledge of higher mathematics is assumed. A basic physics course, although helpful, is not nec- essary for understanding. However, the reader will need to expend some intel- lectual eff ort in grappling with the concepts to come. We realize that not every reader will be interested in the same level of detail. Therefore many (although not all!) of the mathematical details have been placed in appendixes, for those who are interested in more “meat.” Our feeling is that even readers who want to “skip the math” will still fi nd plenty of topics to interest them in our book. So, although we do not expect every reader to understand every single item in the book, we have aimed to provide a stimulating experience for all readers. Interactive Quicktime demonstrations that illustrate some of the concepts in the book can be found at http://press.uchicago.edu/sites/timewarp/. < ix > Acknowledgements We would like to thank Chris Fewster, Larry Ford, David Garfi nkle, Jim Hartle, Bernard Kay, Ken Olum, Amos Ori, David Toomey, Doug Urban, and Alex Vilenkin for useful discussions. We would also like to thank Dave LaPierre and Tim Ouellette for reading the manuscript and providing us with critical comments. Special thanks to Tim Ouellette for ap- plying his considerable editing skills to the manuscript and for his help with the fi gures. Our initial editor at the University of Chicago Press, Jennifer How- ard, gave us constant enthusiastic support during the early stages of this work. Finally, we wish to thank our present editors, Christie Henry, Abby Collier, and especially Mary Gehl, for all their help in turning this manuscript into an actual book. Allen would like to thank his former student, and later colleague, Adel An- tippa, for dragging him in 1970 into what proved to be a stimulating collabora- tive study of the possible physics of tachyons. Adel’s student, now Professor Louis Marchldon, also made important contributions to this work. This laid a foundation for Allen’s renewed interest a quarter of a century later in the physics of superluminal travel and time machines, when interesting new de- velopments began to occur. Allen would also like to extend a special acknowl- edgment to Mrs. Gayle Grant, the secretary of the Physics and Astronomy De- partment at Tufts. Over a number of years, Gayle’s effi ciency and dependability have contributed in countless ways to all aspects of Allen’s professional career, including those connected with this book. Perhaps even more important, her unfailing cheerful friendliness, to faculty and students alike, was an important factor in making the Physics Department a very pleasant place to work. Tom would like to thank the National Science Foundation for partial sup- port under the grant PHY-0968805. < xi > 1 Introduction s humans, we have always been beck- A oned by faraway times and places. Ever since man realized what the stars were, we have wondered whether we would ever be able to travel to them. Such thoughts have provided fertile ground over the years for science fi ction writers seeking interesting plotlines. But the vast distances separating astronomical objects forced authors to invent various imaginary devices that would allow their characters to travel at speeds greater than the speed of light. (The speed of light in empty space, generally denoted as c by physicists, is 186,000 miles/second.) To give you an idea of the enor- mous distances between the stars, let’s start with a few facts. The nearest star, Proxima Centauri (in the Alpha Centauri star system) is about 4 light-years away. A light-year is the distance that light travels in a year, about 6 trillion miles. So the nearest star is about 24 trillion miles away. It would take a beam of light traveling 186,000 miles per second, or a radio message, which would travel at the same speed, 4 years to get there. On an even greater scale, the distance across our Milky Way galaxy is ap- proximately 100,000 light-years. Our nearby neighbor galaxy, Andromeda, is about 2,000,000 light-years away. With present technology, it would take some tens of thousands of years just to send a probe, traveling at a speed far less than c, to the nearest star. It’s not surprising then that science fi ction writers have long imagined some sort of “shortcut” between the stars involving travel faster than the speed of light. Otherwise it is diffi cult to see how one could have the kinds of “federations” or “galactic empires” that are so prominent in science fi ction. Without shortcuts, the universe is a very big place. And what about time, that most mysterious feature of the universe? Why is the past diff erent from the future? Why can we remember the past and not the future? Is it possible that the past and future are “places” that can be visited, just like other regions of space? If so, how could we do it? < 1 > 2 < Chapter 1 This book examines the possibility of time travel and of space travel at speeds exceeding the speed of light, in light of physics research conducted during the last twenty years or so. The ideas of faster-than-light travel and time travel have long existed in popular imagination. What you may not know is that some physicists study these concepts very seriously—not just as a “what might someday be possible” question, but also as a “what can we learn from such studies about basic physics” question. Science fi ction television and movie series, such as Star Trek, contain many fi ctional examples of faster-than-light travel.