Einstein's Apple: Homogeneous Einstein Fields PDF
Preview Einstein's Apple: Homogeneous Einstein Fields
9333_9789814630078_tp.indd 1 23/12/14 9:34 am May2,2013 14:6 BC:8831-ProbabilityandStatisticalTheory PST˙ws TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk World Scientific 9333_9789814630078_tp.indd 2 23/12/14 9:34 am Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. EINSTEIN’S APPLE Homogeneous Einstein Fields Copyright © 2015 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 978-981-4630-07-8 In-house Editor: Christopher Teo Typeset by Stallion Press Email: [email protected] Printed in Singapore Christopher - Einstein's Apple.indd 1 19/12/2014 3:54:39 PM January6,2015 11:37 Einstein’sApple 9inx6in b2002 page-5 This work is dedicated to Alex Harvey who inspired our effort to put all the pieces together. May2,2013 14:6 BC:8831-ProbabilityandStatisticalTheory PST˙ws TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk January6,2015 11:37 Einstein’sApple 9inx6in b2002 page-7 PREFACE The role of homogeneous gravitational fields in the formulation of the equivalence principle and in the foundation of Einstein’s theory of gravita- tion is well known. However, the original treatment of these concepts was done in terms of Newtonian gravity and for small velocities. We believe, therefore, that it is necessaryto treathomogeneousfields relativisticallyin Einstein’s theory of gravitation. In this book we discuss how this can be done for manifolds with simply transitive isometry groups and we mention possible applications. Since the results presented here are far from complete, we are aware of the preliminarycharacterofourinvestigation. What wehavetriedto dois to study the concept of homogeneous fields in Riemannian manifolds from different points of view in an exploratory spirit. This leads to a certain amount of repetition in the different chapters which we hope the reader will excuse. Included are some straightforwardcalculations that we would expect to readily find in the literature but appear to be confined to papers in obscure forms or are not in the common bibliographies and texts. We would like to acknowledge the assistance and support that we have have received from various individuals during the gestation of this work. We would especially like to thank Alex Harvey, Friedrich Hehl, Malcolm MacCallum, Istv´an Ozsva´th, and Andrzej Trautman. Jie Zhao’s contribu- tions, published and unpublished, have been gratefully appreciated. Peter Bergmannshouldbementionedforhislifelongeffortstoclarifythecontent of Einstein’s Theory of Gravitation. New York University provided the facilities where most of the work was carried out. vii May2,2013 14:6 BC:8831-ProbabilityandStatisticalTheory PST˙ws TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk January6,2015 11:37 Einstein’sApple 9inx6in b2002 page-9 CONTENTS Preface vii Table of Contents ix List of Figures xi 0. “The Happiest Thought of My Life” 1 1. Accelerated Frames 13 2. Torsion and Telemotion 28 3. InertialandGravitationalFieldsinMinkowski Spacetime 38 4. The Notion of Torsion 47 5. Homogeneous Fields on Two-dimensional Riemannian Manifolds 60 6. Homogeneous Vector Fields in N-dimensions 79 7. Homogeneous Fields on Three-dimensional Spacetimes: Elementary Cases 94 8. Proper Lorentz Transformations 111 9. Limits of Spacetimes 136 10. HomogeneousFieldsinMinkowskiSpacetimes 162 11. Euclidean Three-dimensional Spaces 182 12. Homogeneous Fields in Arbitrary Dimension 208 13. Summary 225 Appendix A. Basic Concepts 229 Appendix B. A Non-trivial Global Frame Bundle 240 Appendix C. Geodesics of the Poincar´e Half-Plane 244 ix
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