The Mathematical Theory of Cosmic Strings

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String theory

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THE MATHEMATICAL THEORY OF COSMIC STRINGSCOSMIC STRINGS IN THE WIRE APPROXIMATION

Series in High Energy Physics, Cosmology and GravitationOther books in the series ElectronPositron Physics at the Z M G Green, S L Lloyd, P N Ratoff and D R Ward Non-accelerator Particle Physics Paperback edition H V Klapdor-Kleingrothaus and A Staudt Ideas and Methods of Supersymmetry and Supergravity or A Walk Through Superspace Revised edition I L Buchbinder and S M Kuzenko Pulsars as Astrophysical Laboratories for Nuclear and Particle Physics F Weber Classical and Quantum Black Holes Edited by P Fr , V Gorini, G Magli and U Moschella e Particle Astrophysics Revised paperback edition H V Klapdor-Kleingrothaus and K Zuber The World in Eleven Dimensions Supergravity, Supermembranes and M-Theory Edited by M J Duff Gravitational Waves Edited by I Ciufolini, V Gorini, U Moschella and P Fr e Modern Cosmology Edited by S Bonometto, V Gorini and U Moschella Geometry and Physics of Branes Edited by U Bruzzo, V Gorini and U Moschella The Galactic Black Hole Lectures on General Relativity and Astrophysics Edited by H Falcke and F W Hehl

THE MATHEMATICAL THEORY OF COSMIC STRINGSCOSMIC STRINGS IN THE WIRE APPROXIMATION

Malcolm R AndersonDepartment of Mathematics, Universiti Brunei, Darussalam

I NSTITUTE OF P HYSICS P UBLISHING B RISTOL AND P HILADELPHIA

c IOP Publishing Ltd 2003 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 permission of the publisher. Multiple copying is permitted in accordance with the terms of licences issued by the Copyright Licensing Agency under the terms of its agreement with Universities UK (UUK). British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. ISBN 0 7503 0160 0 Library of Congress Cataloging-in-Publication Data are available

Commissioning Editor: James Revill Production Editor: Simon Laurenson Production Control: Sarah Plenty Cover Design: Victoria Le Billon Marketing: Nicola Newey and Verity Cooke Published by Institute of Physics Publishing, wholly owned by The Institute of Physics, London Institute of Physics Publishing, Dirac House, Temple Back, Bristol BS1 6BE, UK US Ofce: Institute of Physics Publishing, The Public Ledger Building, Suite 929, 150 South Independence Mall West, Philadelphia, PA 19106, USAA Typeset in LTEX 2 by Text 2 Text, Torquay, Devon Printed in the UK by MPG Books Ltd, Bodmin, Cornwall

Contents

Introduction 1 Cosmic strings and broken gauge symmetries 1.1 Electromagnetism as a local gauge theory 1.2 Electroweak unication 1.3 The NielsenOlesen vortex string 1.4 Strings as relics of the Big Bang 1.5 The Nambu action The elements of string dynamics 2.1 Describing a zero-thickness cosmic string 2.2 The equation of motion 2.3 Gauge conditions, periodicity and causal structure 2.4 Conservation laws in symmetric spacetimes 2.5 Invariant length 2.6 Cusps and curvature singularities 2.7 Intercommuting and kinks String dynamics in at space 3.1 The aligned standard gauge 3.2 The GGRT gauge 3.3 Conservation laws in at space 3.4 Initial-value formulation for a string loop 3.5 Periodic solutions in the spinor representation 3.6 The KibbleTurok sphere and cusps and kinks in at space 3.7 Field reconnection at a cusp 3.8 Self-intersection of a string loop 3.9 Secular evolution of a string loop A bestiary of exact solutions 4.1 Innite strings 4.1.1 The innite straight string 4.1.2 Travelling-wave solutions 4.1.3 Strings with paired kinks 4.1.4 Helical strings

ix 1 3 8 15 24 27 35 35 38 41 44 48 49 54 59 59 61 63 68 70 73 80 85 92 99 99 99 100 102 103

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3

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vi 4.2

ContentsSome simple planar loops 4.2.1 The collapsing circular loop 4.2.2 The doubled rotating rod 4.2.3 The degenerate kinked cuspless loop 4.2.4 Cats-eye strings Balloon strings Harmonic loop solutions 4.4.1 Loops with one harmonic 4.4.2 Loops with two unmixed harmonics 4.4.3 Loops with two mixed harmonics 4.4.4 Loops with three or more harmonics Stationary rotating solutions Three toy solutions 4.6.1 The teardrop string 4.6.2 The cardioid string 4.6.3 The gure-of-eight string 105 105 106 107 108 112 114 114 117 122 127 130 135 135 137 141 144 144 145 147 152 153 157 159 167 170 177 181 182 185 189 191 196 197 199 202 204 208 211 211 213 219 219

4.3 4.4

4.5 4.6

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String dynamics in non-at backgrounds 5.1 Strings in RobertsonWalker spacetimes 5.1.1 Straight string solutions 5.1.2 Ring solutions 5.2 Strings near a Schwarzschild black hole 5.2.1 Ring solutions 5.2.2 Static equilibrium solutions 5.3 Scattering and capture of a straight string by a Schwarzschild hole 5.4 Ring solutions in the Kerr metric 5.5 Static equilibrium congurations in the Kerr metric 5.6 Strings in plane-fronted-wave spacetimes Cosmic strings in the weak-eld approximation 6.1 The weak-eld formalism 6.2 Cusps in the weak-eld approximation 6.3 Kinks in the weak-eld approximation 6.4 Radiation of gravitational energy from a loop 6.5 Calculations of radiated power 6.5.1 Power from cuspless loops 6.5.2 Power from the VachaspatiVilenkin loops 6.5.3 Power from the p/q harmonic solutions 6.6 Power radiated by a helical string 6.7 Radiation from long strings 6.8 Radiation of linear and angular momentum 6.8.1 Linear momentum 6.8.2 Angular momentum 6.9 Radiative efciencies from piecewise-linear loops 6.9.1 The piecewise-linear approximation

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Contents6.9.2 A minimum radiative efciency? 6.10 The eld of a collapsing circular loop 6.11 The back-reaction problem 6.11.1 General features of the problem 6.11.2 Self-acceleration of a cosmic string 6.11.3 Back-reaction and cusp displacement 6.11.4 Numerical results

vii 223 226 231 231 234 240 242

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The gravitational eld of an innite straight string 7.1 The metric due to an innite straight string 7.2 Properties of the straight-string metric 7.3 The GerochTraschen critique 7.4 Is the straight-string metric unstable to changes in the equation of state? 7.5 A distributional description of the straight-string metric 7.6 The self-force on a massive particle near a straight string 7.7 The straight-string metric in asymptotically-at form

246 246 250 252 255 259 263 267

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Multiple straight strings and closed timelike curves 8.1 Straight strings and 2 + 1 gravity 8.2 Boosts and rotations of systems of straight strings 8.3 The Gott construction 8.4 String holonomy and closed timelike curves 8.5 The LetelierGaltsov spacetime

271 271 273 274 278 282

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Other exact string metrics 9.1 Strings and travelling waves 9.2 Strings from axisymmetric spacetimes 9.2.1 Strings in a RobertsonWalker universe 9.2.2 A string through a Schwarzschild black hole 9.2.3 Strings coupled to a cosmological constant 9.3 Strings in radiating cylindrical spacetimes 9.3.1 The cylindrical formalism 9.3.2 Separable solutions 9.3.3 Strings in closed universes 9.3.4 Radiating strings from axisymmetric spacetimes 9.3.5 EinsteinRosen soliton waves 9.3.6 Two-mode soliton solutions 9.4 Snapping cosmic strings 9.4.1 Snapping strings in at spacetimes 9.4.2 Other spacetimes containing snapping strings

286 286 291 292 297 301 303 303 305 307 310 316 321 324 324 329

viii

Contents332 334 340 343 346 346 349 352 355 359 367

10 Strong-eld effects of zero-thickness strings 10.1 Spatial geometry outside a stationary loop 10.2 Black-hole formation from a collapsing loop 10.3 Properties of the near gravitational eld of a cosmic string 10.4 A 3 + 1 split of the metric near a cosmic string 10.4.1 General formalism 10.4.2 Some sample near-eld expansions 10.4.3 Series solutions of the near-eld vacuum Einstein equations 10.4.4 Distributional stressenergy of the world sheet Bibliography Index

Introduction

The existence of cosmic strings was rst proposed in 1976 by Tom Kibble, who drew on the theory of line vortices in superconductors to predict the formation of similar structures in the Universe at large as it expanded and cooled during the early phases of the Big Bang. The critical assumption is that the strong and electroweak forces were rst isolated by a symmetry-breaking phase transition which converted the energy of the Higgs eld into the masses of fermions and vector bosons. Under certain conditions, it is possible that some of the Higgs eld energy remained in thin tubes which stretched across the early Universe. These are cosmic strings. The masses and dimensions of cosmic strings are largely determined by the energy scale at which the relevant phase transition occurred. The grand unication (GUT) energy scale is at present estimated to be about 1015 GeV, which indicates that the GUT phase transition took place some 1037 1035 s after the Big Bang, when the temperature of the Universe was of the order of 1028 K. The thickness of a cosmic string is typically comparable to the Compton wavelength of a particle with GUT mass or about 1029 cm. This distance is so much smaller than the length scales important to astrophysics and cosmology that cosmic strings are usually idealized to have zero thickness. The mass per unit length of such a string, conventionally denoted , is proportional to the square of the energy scale, and in the GUT case has a value of about 1021 g cm1 . There is no restriction on the length of a cosmic string, although in the simplest theories a string can have no free ends and so must either be innite or form a closed loop. A GUT string long enough to cross the observable Universe would have a mass within the horizon of about 1016 M , which is no greater than the mass of a large cluster of galaxies. Interest in cosmic strings intensied in 198081, when Yakov Zeldovich and

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