Godel Meets Einstein : Time Travel in the Godel Universe

Godel Meets Einstein Time Travel in the Godel Universe

PALLE YOURGRAU

OPEN COURT Chicago and La Salle, Illinois

To order books from Open Court, c,.'1111-800-815-2280. The front cover photograph of Gode! and Einstein is used by courtesy of Richard Arens. The back cover photograph of Palle Yourgrau at Gi1del's grave is by M.K. Sullivan.

Open Court Publishing Company is a division of Carus Publishing Company. Copyright © 1999 by Carns Publishing Company hrst printing 1999 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher, Open Court Publishing Company, a division of Carns Publishing Company, 315 fifth Street, P.O. Box 300, Peru, Illinois 61354-0300. Printed and bound in the United States of America.

Library of Congress Cataloging-in-Publication Data Yourgrau, Palle Giidelmeets Einstein: time travel in the Gbdcluniverse/i'alle Yourgrau. p. cm. Rev. ed. of: The disappearance of time. Includes bibliographical rderences (p.) and index. ISBN 0-8126-9408-2 (pbk. : alk.) 1. Giidd, Kurt-Contributions in concept of time. 2. Time-History-20th century. 3. Einstein, Albert, 1879-1955. I. Yourgrau, Palle. Disappearance of time. II. Title. BD638.Y68 1999 99-048706 115' .092-dc21

If the world is beautiful, and its maker good, clearly he had his eye on the eternal. PLATO

... projecting directly from the developed parts of mathematics to the whole world [Codel] concludes that it is also beautiful and perfect. HAOWANG

Contents

Preface to the NelV Expanded Edition (1999) Preface to the First Edition (1991)

Chapter 1 The Reception of G6del's Results Chapter 2 G6del's Idealism

IX

XXI

1

17

Chapter 3 Time Travel and the G6del Universe 37

Chapter 4 Not Everything Can Be Relativized

51

Chapter 5 From Kant to Star Trek: A Philosopher's Guide to Time Travel

69

Chapter 6 Formalization and Representation

123

V1Il

Contents

Chapter 7 Being and Time

169

Appendix A Brouwer and the 'Revolution': The Philosophical Background of 'Temporal Mathematics'

205

Appendix B Zeno's Revenge: On the Mathematical Dogma of Infinite Divisibility

213

Bibliography

233

Index

247

Preface to the New Expanded Edition (1999)

At mid-century, Kurt G6del and Albert Einstein, two of the greatest thinkers not just of the century but of the millennium, were colleagues at the Institute for Advanced Study in Princeton, New Jersey, and also the closest of friends. What, one wonders, would have happened if G6del had taken the opportunity to cast a mathematical eye on Einstein's Theory of Relativity? Would not something amazing have resulted? Well, fate sometimes smiles on us and as it happens G6del did in fact turn his attention to Einstein's relativistic cosmology and the results were, predictably, astonishing. Within short compass G6del discovered new and entirely unexpected solutions to the field equations of general relativity, bringing to light world models with disconcerting properties. In particular, he demonstrated the possibility, consistent with the laws of general relativity, of 'Rotating Universes', known today as 'G6del Universes', in which, provably, there exist closed timelike world lines that permit, in a strict sense, time travel. In 1949, then, G6del became the first thinker in the history of science to demonstrate rigorously the genuine possibility-consistent with the established laws of nature-of time travel. The mid-century meeting of minds of G6del and Einstein gave birth, then, to a beautiful child of the scientific imagination. Gbdel, already in his twenties, had begun to establish himself as, in one colleague's words, "the greatest logician since Aristotle"J. Indeed, as the twentieth century draws to a close, it has become increasingly clear that Gbdel's famous IncompleteI. This colleague went on to assert that this description constituted all ullderevaluatioll of Giidd I IX

x

Preface to the New Expanded Edition (1999)

ness Theorem for mathematical logic stands alongside Heisenberg's Uncertainty Principle and Einstein's Theory of Relativity as one the great mathematical achievements of this-or any other-century 2. Indeed, these three milestones of twentieth century science have much in common. Although all three represent high water marks in the most rigorous of mathematical sciences, each constitutes at the same time a fountain of philosophical inspiration. In particular, although each is established by formal methods, each demonstrates, in its own peculiar way, a kind of limitation in principle of the relevant formal science. Heisenberg sets limits to our simultaneous knowledge of the position and momentum of the fundamental particles. Einstein, in turn, sets a limit to the speed of light and indeed to the velocity of any information-bearing signal in the universe. And G6del establishes limits in the ability of any strictly formal, axiomatic mathematical system-in particular, of any computer programto capture not only all mathematical truth, but even the totality of truths of arithmetic. 3 It is a further, striking fact that all three thinkers draw ontological consequences, about the nature of reality, from what are in effect epistemic premises. Philosophically speaking, however, although a handful of G6del's essays advancing his 'Platonistic' philosophy of mathematics have achieved, if not acceptance, then at least a kind of notoriety, it is rarely recognized that his writings on Einstein, in which he takes on one of the greatest and most elusive of philosophical questions-the reality of time-bear the unmistakable stamp of philosophical greatness. His argument proceeds in stages. 1. He argues forcefully that in the Special Theory of Relativity (STR) the relativity of the 'now' (of intuitive time) to an inertial frame 4 implies the relativity of existence. 5 But the latter is impossible. 6 Hence, if the STR is true, (intuitive) time disap2. When in 1952 Godel was granted an honorary doctorate by Harvard University, the citation by the distinguished logician, W.V. Quine, described Giidcl as "the discoverer of the most significant mathematical result of the century." Godel, in a letter to his mother, explained that the citation should not be taken to mean that he is the greatest mathematician of the century, but rather that the phrase, "most significant" means here, "of the greatest General Interest outside of mathematics". 3. That is, of so-called 'higher arithmetic' or number theOlY. 4. It's an axiom of the STR that no inertial frame is 'privileged'. 5. " ... [TJhe idea of an objective lapse of time (whose essence is that only the present really exists) ... " (Giidel1949a, p. 202, Note 4). 6. "The concept of existence ... cannot be relativized without destroying its meaning completely" (Gt'idelI949a, p. 203, Note 5, in Fcferman et aI., Vol. II).

Preface to the New Expanded Edition (1999)

Xl

pears. 2. In the General Theory of Relativity (GTR), however, where matter affects spacetime curvature, some reference frames are 'privileged'-namely those whose motion "follows the mean motion of matter" (G6del 1949a, p. 204). In our universe, these privileged reference frames can be systematically coordinated to determine a single, objective, 'cosmic time'. Time reappears. 3. a) G6del discovered, however, world models for the GTR--the Rotating or G6del Universes (GU's)-in which, provably, cosmic time disappears. b) Further, in certain nonexpanding GU's, there exist closed timelike world lines, permitting time travel. But if you can revisit the past, it never 'passed' from existence in the first place. So, once again, time disappears. 4. But the GU is a mere possible world. What about the actual world? By a daring 'modal' argument (from the possible to the actual), G6del reasons as follows. Our world and the GU differ only in the global distribution of matter and motion. The two universes are described by the same fundamental laws of nature and, conceivably, provide for observers the same experiences (as of) time. If time is an illusion in the one world, therefore (as it provably is), it must also be an illusion in the other? So, in the last analysis, time disappears even in the actual world. G6del's reasoning in 1.-4. is surely clear evidence of an extraordinary philosophical intelligence; yet his dramatic achievements in relativistic cosmology and the philosophy of time have themselves, one might say, 'disappeared' trom view. Cosmologists, it is true, when G6del first announced the existence of his new world models, were intrigued and were spurred on to discover other temporally anomalous world models for general relativity. G6del had succeeded, at a minimum, in waking up general relativists to the fact that Einstein's theory by no means ruled out the possible existence of world models containing temporal and causal anomalies. Interest in the G6del Universe as such, however, was primarily confined to demonstrations that the actual world is not such a universe,8 as well as to queries as to whether the GU was after all a physically realistic solution to Einstein's Field Equations, as

7. "A lapse of time ... would have to be f