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100 YEARS of RELATIVITY Space-Time Structure: Einstein and Beyond Copyright © 2005 by World Scientific Publishing Co This page intentionally left blank 100 Years Relativity Space-Time Structure: Einstein and Beyond This page intentionally left blank KMears Relativity Space-Time Structure: Einstein and Beyond editor Abhay Ashtekar Institute for Gravitational Physics and Geometry, Pennsylvania State University, USA "X Vfe World Scientific \ 11B^ NEW JERSEY' • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONHONG KONG • TAIPEI • CHENNAI 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. Cover Art by Dr. Cliff Pickover, Pickover.Com 100 YEARS OF RELATIVITY Space-Time Structure: Einstein and Beyond Copyright © 2005 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 981-256-394-6 Printed in Singapore. October 12, 2005 14:40 WSPC/Trim Size: 9in x 6in for Review Volume preface PREFACE The goal of this volume is to describe how our understanding of space-time structure has evolved since Einstein's path-breaking 1905 paper on special relativity and how it might further evolve in the next century. Preoccupation with notions of Space (the Heavens) and Time (the Beginning, the Change and the End) can be traced back to at least 2500 years ago. Early thinkers from Lao Tsu in China and Gautama Buddha in India to Aristotle in Greece discussed these issues at length. Over cen- turies, the essence of Aristotle's commentaries crystallized in the Western consciousness, providing us with mental images that we commonly use. We think of space as a three-dimensional continuum which envelops us. We think of time as flowing serenely, all by itself, unaffected by forces in the physical universe. Together, they provide a stage on which the drama of interactions unfolds. The actors are everything else | stars and planets, ra- diation and matter, you and me. In Newton's hands, these ideas evolved and acquired a mathematically precise form. In his masterpiece, the Principia, Newton spelled out properties of space and the absolute nature of time. The Principia proved to be an intellectual tour de force that advanced Science in an unprecedented fashion. Because of its magnificent success, the notions of space and time which lie at its foundations were soon taken to be obvious and self-evident. They constituted the pillars on which physics rested for over two centuries. It was young Einstein who overturned those notions in his paper on special relativity, published on 26th September 1905 in Annalen der Physik (Leipzig). Lorentz and Poincar´e had discovered many of the essential math- ematical underpinnings of that theory. However, it was Einstein, and Ein- stein alone, who discovered the key physical fact | the Newtonian notion of absolute simultaneity is physically incorrect. Lorentz transformations of time intervals and spatial lengths are not just convenient mathematical ways of reconciling experimental findings. They are physical facts. Newton's assertion that the time interval between events is absolute and observer- independent fails in the real world. The Galilean formula for transformation of spatial distances between two events is physically incorrect. What seemed obvious and self-evident for over two centuries is neither. Thus, the model v October 12, 2005 14:40 WSPC/Trim Size: 9in x 6in for Review Volume preface vi A. Ashtekar of space-time that emerged from special relativity was very different from that proposed in the Principia. Space-time structure of special relativity has numerous radical ramifica- tions | such as the twin \paradox" and the celebrated relation E = Mc2 between mass and energy | that are now well known to all physicists. However, as in the Principia, space-time continues to remain a backdrop, an inert arena for dynamics, which cannot be acted upon. This view was toppled in 1915, again by Einstein, through his discovery of general rela- tivity. The development of non-Euclidean geometries had led to the idea, expounded most eloquently by Bernhard Riemann in 1854, that the geom- etry of physical space may not obey Euclid's axioms | it may be curved due to the presence of matter in the universe. Karl Schwarzschild even tried to measure this curvature as early as 1900. But these ideas had remained speculative. General relativity provided both a comprehensive conceptual foundation and a precise mathematical framework for their realization. In general relativity, gravitational force is not like any other force of Nature; it is encoded in the very geometry of space-time. Curvature of space-time provides a direct measure of its strength. Consequently, space- time is not an inert entity. It acts on matter and can be acted upon. As John Wheeler puts it: Matter tells space-time how to bend and space-time tells matter how to move. There are no longer any spectators in the cosmic dance, nor a backdrop on which things happen. The stage itself joins the troupe of actors. This is a profound paradigm shift. Since all physical systems reside in space and time, this shift shook the very foundations of natural philosophy. It has taken decades for physicists to come to grips with the numerous ramifications of this shift and philosophers to come to terms with the new vision of reality that grew out of it. Einstein was motivated by two seemingly simple observations. First, as Galileo demonstrated through his famous experiments at the leaning tower of Pisa, the effect of gravity is universal: all bodies fall the same way if the only force acting on them is gravitational. This is a direct consequence of the equality of the inertial and gravitational mass. Second, gravity is always attractive. This is in striking contrast with, say, the electric force where unlike charges attract while like charges repel. As a result, while one can easily create regions in which the electric field vanishes, one cannot build gravity shields. Thus, gravity is omnipresent and nondiscriminating; it is everywhere and acts on everything the same way. These two facts make gravity unlike any other fundamental force and suggest that gravity is a manifestation of something deeper and universal. Since space-time is October 12, 2005 14:40 WSPC/Trim Size: 9in x 6in for Review Volume preface Preface vii also omnipresent and the same for all physical systems, Einstein was led to regard gravity not as a force but a manifestation of space-time geometry. Space-time of general relativity is supple, depicted in the popular literature as a rubber sheet, bent by massive bodies. The sun, for example, bends space-time nontrivially. Planets like earth move in this curved geometry. In a precise mathematical sense, they follow the simplest trajectories called geodesics | generalizations of straight lines of the flat geometry of Euclid to the curved geometry of Riemann. Therefore, the space-time trajectory of earth is as \straight" as it could be. When projected into spatial sec- tions defined by sun's rest frame, it appears elliptical from the flat space perspective of Euclid and Newton. The magic of general relativity is that, through elegant mathematics, it transforms these conceptually simple ideas into concrete equations and uses them to make astonishing predictions about the nature of physical re- ality. It predicts that clocks should tick faster on Mont Blanc than in Nice. Galactic nuclei should act as giant gravitational lenses and provide spectac- ular, multiple images of distant quasars. Two neutron stars orbiting around each other must lose energy through ripples in the curvature of space-time caused by their motion and spiral inward in an ever tightening embrace. Over the last thirty years, precise measurements have been performed to test if these and other even more exotic predictions are correct. Each time, general relativity has triumphed. The accuracy of some of these observa- tions exceeds that of the legendary tests of quantum electrodynamics. This combination of conceptual depth, mathematical elegance and observational success is unprecedented. This is why general relativity is widely regarded as the most sublime of all scientific creations. Perhaps the most dramatic ramifications of the dynamical nature of geometry are gravitational waves, black holes and the big-bang.a aHistory has a tinge of irony | Einstein had difficulty with all three! Like all his con- temporaries, he believed in a static universe. Because he could not find a static solution to his original field equations, in 1917 he introduced a cosmological constant term whose repulsive effect could balance the gravitational attraction, leading to a large scale time independence. His belief in the static universe was so strong that when Alexander Fried- mann discovered in 1922 that the original field equations admit a cosmological solution depicting an expanding universe, for a whole year, Einstein thought Friedmann had made an error. It was only in 1931, after Hubble's discovery that the universe is in fact dy- namical and expanding that Einstein abandoned the extra term and fully embraced the expanding universe. The story of gravitational waves is even more surprising. In 1936 Einstein and Nathan Rosen submitted a paper to Physical Review in which they argued that Einstein's 1917 weak field analysis was misleading and in fact gravitational waves do not exist in full general relativity.
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