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Fundamental Theories of Physics 193 Serguei Krasnikov Back-in-Time and Faster-than-Light Travel in General Relativity Fundamental Theories of Physics Volume 193 Series editors Henk van Beijeren, Utrecht, The Netherlands Philippe Blanchard, Bielefeld, Germany Paul Busch, York, United Kingdom Bob Coecke, Oxford, United Kingdom Dennis Dieks, Utrecht, The Netherlands Bianca Dittrich, Waterloo, Canada Detlef Dürr, Munich, Germany Ruth Durrer, Geneva, Switzerland Roman Frigg, London, United Kingdom Christopher Fuchs, Boston, USA Giancarlo Ghirardi, Trieste, Italy Domenico J. W. Giulini, Bremen, Germany Gregg Jaeger, Boston, USA Claus Kiefer, Cologne, Germany Nicolaas P. Landsman, Nijmegen, The Netherlands Christian Maes, Leuven, Belgium Mio Murao, Bunkyo-ku, Tokyo, Japan Hermann Nicolai, Potsdam, Germany Vesselin Petkov, Montreal, Canada Laura Ruetsche, Ann Arbor, USA Mairi Sakellariadou, London, UK Alwyn van der Merwe, Denver, USA Rainer Verch, Leipzig, Germany Reinhard Werner, Hannover, Germany Christian Wüthrich, Geneva, Switzerland Lai-Sang Young, New York City, USA The international monograph series “Fundamental Theories of Physics” aims to stretch the boundaries of mainstream physics by clarifying and developing the theoretical and conceptual framework of physics and by applying it to a wide range of interdisciplinary scientific fields. Original contributions in well-established fields such as Quantum Physics, Relativity Theory, Cosmology, Quantum Field Theory, Statistical Mechanics and Nonlinear Dynamics are welcome. The series also provides a forum for non-conventional approaches to these fields. Publications should present new and promising ideas, with prospects for their further development, and carefully show how they connect to conventional views of the topic. Although the aim of this series is to go beyond established mainstream physics, a high profile and open-minded Editorial Board will evaluate all contributions carefully to ensure a high scientific standard. More information about this series at http://www.springer.com/series/6001 Serguei Krasnikov Back-in-Time and Faster-than-Light Travel in General Relativity 123 Serguei Krasnikov St. Petersburg Russia ISSN 0168-1222 ISSN 2365-6425 (electronic) Fundamental Theories of Physics ISBN 978-3-319-72753-0 ISBN 978-3-319-72754-7 (eBook) https://doi.org/10.1007/978-3-319-72754-7 Library of Congress Control Number: 2017962017 © Springer International Publishing AG, part of Springer Nature 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by the registered company Springer International Publishing AG part of Springer Nature The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Речь идет о том, как поступать с задачей, которая решения не имеет. Это глубоко принципиальный вопрос […] К. Х. Хунта в [189]1 1 “We are speaking of how to deal with a problem that has no solution. This is a matter of deep philosophical principle …” Cristobal Junta in [189]. To my parents Preface Can a cause–effect chain close into a circle? Can a person travel to their past? These questions have been intriguing people for millennia. Until the last century, however, they were discussed only by philosophers and poets.2 An adequate language appeared only in the end of the nineteenth century, when due to the efforts of people like Wells and Poincare the concept was devel- oped of time as an attribute of a four-dimensional object now called Minkowski space. This concept formed the foundation of special relativity—logically, if not historically—and brought up the third, almost equivalent to the first two, question: is it possible to travel faster than light? The next major step in understanding causality3 was made in a few years, when Einstein developed general relativity, according to which the Minkowski space is only an approximation, while the actual large-scale geometry of the universe is much more complex and admits—in principle—different wonders: handles, frac- tures, closed geodesics, etc. As a result, the question of a maximal speed became much harder, even the concept of distance lost its clarity. The problem of time travel also acquired a new aspect: among the above-mentioned wonders, there may be self-intersecting timelike curves. Such a curve would describe an object—a time traveller—returning to the past in a manner that has no analogues in special rela- tivity. The traveller observes no local ‘miracles’: photons still overtake it, people do not walk backward, raindrops fall down from clouds, etc. Nevertheless, at some moment they meet their younger/older self. To dismiss the possibility of such time trips is much more difficult than those in the Minkowski space. Einstein confessed 2Oedipus receives (via an oracle) a signal from the future, which compels him to flee from Corinth. This flight is responsible for his meeting (and, consequently, killing) Laius. But it is the message from the oracle concerning this killing that prompted Oedipus to leave Corinth (where he would have never meet Laius). The story comes full circle making up what is known today as a ‘bootstrap paradox’, see Chap. 6. 3 The importance of that step went far beyond physics. It is worth mentioning, for example, that modern philosophers of time use a general relativity based language [38]. ix x Preface that this problem disturbed him already at the time of the building up of the general theory of relativity4 without his having succeeded in clarifying it [40]. In a next few decades, little progress was made. The finding by Gödel [66] of a cosmological model in which causality breaks down in every point somehow had no developments. And the pithy discussion of tachyons concerned only causality in Minkowski space. True, at that time the concept of wormhole gained popularity. Misner and Wheeler suggested that what we think are charges are actually wormhole mouths [134]. This idea, however, was not turned into a realistic model. At the same time, the specific wormholes proposed by Ellis [44] and Bronnikov [18] required the matter source to be quite exotic (which is unavoidable, as we understand now). As a result, the interest in wormholes waned. The situation changed drastically after the paper by Morris and Thorne had been published where they noted that the existence of wormholes would apparently imply also the existence of closed timelike curves [136]. This idea aroused con- siderable interest and gave rise to a wave of publications. In a few years, however, it became clear—somewhat unexpectedly, it seems—that there are no obvious arguments against the possibility of time travel. This ended the period of facile attacks leaving us with the understanding that 1. the problem was harder than it looked; 2. the formation of a time machine may be unstable; 3. both time and faster-than-light journeys (the latter was represented by a flight through a wormhole) require, as a source of the spacetime curvature, matter with negative (in a certain frame) energy density. The results obtained by the early 1990s are summed up—with an emphasis on wormholes—in [172]. The next period was characterized by a shift of accents. In particular, the wormholes, time machines and—after the publication of Alcubierre’s paper [3]— superluminal motion, began to be studied on equal footing with other relativistic phenomena (singularities, say), and not as curiosities, whose impossibility, though obvious, still is not convincingly proven, for some reason. The new approach turned out to be more productive and in the last 20 years a number of interesting results have been obtained. The time seems right to gather them into one mono- graph and thus to spare those interested in the subject from having to work through numerous original papers. That is how this book came about (see also [187]). Not being a review, it is not intended to be complete: to keep the size of the book reasonable without sacrificing thoroughness, some topics were not included. The criteria of rejection were sub- jective; it is worth mentioning only that among the omitted topics there are important (e.g. Gott’s time machine), fashionable (e.g. wormholes in the ‘thin shell approximation’) and highly controversial (e.g. Fomalont and Kopeikin’s mea- surement of the speed of gravity) ones. 4 Maybe, in fact, even earlier [41]. Preface xi Though some of its parts must be understandable to a reader unfamiliar with mathematics, this book is by no means popular. To understand it entirely, including the proofs, a reader should be acquainted with basic notions of general topology (such as connectedness, compactness, etc.) and differential geometry (manifolds, tensors, etc.). It will be helpful also, if the reader is familiar with the basics of relativity. On the other hand, some fundamentals are recapitulated in the first four sections of Chap. 1. I acknowledge the many people who have helped this book to appear.

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