Einstein’s Space-Time An Introduction to Special and General Relativity Rafael Ferraro Universidad de Buenos Aires Consejo Nacional de Investigaciones Científicas y Técnicas (República Argentina) Rafael Ferraro Universidad de Buenos Aires Consejo Nacional de Investigaciones Científicas y Técnicas (República Argentina) Email: [email protected] Library of Congress Control Number: 2007923177 ISBN 978-0-387-69946-2 eISBN 978-0-387-69947-9 © 2007 Springer Science + Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. 987654321 springer.com Preface Between 1994 and 1999, I had the pleasure of lecturing Special and General Relativity in the Facultad de Ciencias Exactas y Naturales of the Universidad de Buenos Aires. These lectures were targeted to undergraduate and graduate students of Physics. However, it is increasingly apparent that interest in Relativity extends beyond these academic circles. Because of this reason, this book intends to become useful to students of related disciplines and to other readers interested in Einstein’s work, who will be able to incorporate entirely the fundamental ideas of Relativity starting from the very basic concepts of Physics. To understand the Theory of Relativity it is necessary to give up our intuitive notions of space and time, i.e., the notions used in our daily relation with the world. These classical notions of space and time are also the foundations of Newtonian mechanics, which dominated Physics for over two centuries until they clashed with Maxwell’s electromagnetism. Classical physics assumed that space is immutable and its geometry obeys the Euclidean postulates. Furthermore, distances and time intervals are believed invariant, i.e., independent of the state of motion. Both preconceptions about the nature of space and time rely firmly on our daily experience, in such a way that the classical notions are imprinted in our thought with the status of “true.” Therefore, we tend to resist abandoning the classical notions of space and time, and to replace them with other notions that are not evident in the phenomena observed in our daily life. This state of affairs can take the student of Relativity to a mode of thinking vitiated by the coexistence of old and new conceptions of space and time, which would lead to several perplexities and paradoxes. The historical approach of this text attempts to guide the reader along the same road followed by Physics between the seventeenth and twentieth centuries, thus reproducing the intellectual process that led to the relativistic way of conceiving the space-time. After this journey, the reader will become more receptive to the consequences of the change of paradigm. Chapters 1–7 contain a complete course on Special Relativity that can be adapted to an introductory level by means of a suitable choice of issues. In this sense, the text is conceived to allow its reading in several levels, since some subjects that could hinder the reading by beginners have been located as Complements inside boxes. The historical road toward Special Relativity is v vi Preface covered in the first two chapters. They begin with the dispute between Leibniz and Newton about the relational or absolute character of space—an issue that paves the way for introducing General Relativity in Chapter 8—and the analysis of the role played by absolute distances and times in the foundations of Newton’s Dynamics. The two mechanical theories of light—corpuscular model and wave theory of luminiferous ether—are developed in detail in Chapter 2, thus arriving at the tensions arisen between Maxwell’s electromagnetic ether theory and the Principle of relativity in the second half of the nineteenth century. The challenge posed by experimental results (Arago, Hoek, Airy, Michelson-Morley, etc.) and the theoretical efforts to solve it (Fresnel, Lorentz, Poincaré, etc.) prepare the reader for the arrival of Einstein’s theory. Chapter 3 introduces the postulates of Special Relativity and the notions of space and time derived from them. Here mathematics is entirely basic, because no differential calculus is needed. The understanding of length contraction, time dilatation, Lorentz transforma- tions, relativistic addition of velocities, etc. only entails elementary concepts of kinematics. This Chapter deals with replacing classical notions of space and time with those notions that emerge from postulating that Maxwell’s laws are valid in any inertial frame. Therefore, the central goal here is to enable a full understanding of the interrelation between length contraction and time dilatation and its consequences. Chapter 3 ends with the transformation of a plane wave, where its connection with the foundations of Quantum Mechanics is emphasized. Chapter 4 develops the geometric properties of Minkowski space- time (invariance of the interval, causal structure, light cone, etc.) together with some advanced topics like Wigner rotation. Chapter 5 teaches the transfor- mations of the electromagnetic field, charge and current densities, potentials, etc. in the context of the ordinary vector language, starting from the preser- vation of the polar and axial characters of electric and magnetic fields, and the essential features of a plane electromagnetic wave. It also contains the fields of moving charges, dipoles, and continuous media. Chapter 6 is devoted to relativistic Dynamics, and explains the changes suffered by Newton’s Dynamics in order to bring their laws into agreement with the Principle of relativity under Lorentz transformations. The reformulation of the Dynamics leads to the “mass- energy equivalence,” so the rich phenomenology in atomic and nuclear physics concerned with this topic is covered. Chapter 7 introduces the four-tensor formu- lation of Special Relativity, and applies it to several subjects that enter into play also in General Relativity: volume and hypersurfaces, energy-momentum tensor of a fluid, electromagnetism, Fermi–Walker transport, etc. Chapter 8 displays General Relativity—the relativistic theory for the gravitational-inertial field—starting from Mach’s criticism of Mechanics as a trigger of Einstein’s thought. Once again, the explanation transits the historical way by focusing on the beginning of the new concepts. This chapter introduces the mathematical tools of the geometric language—covariant derivative, Riemann tensor, etc.—avoiding excesses of mathematical complexity. The Einstein equations for the geometry of space-time are formulated, and their consequences in the weak field approxi- mation are studied to connect with Newtonian gravity. The issue of the number Preface vii of degrees of freedom and constraint equations is tackled, and exemplified in the context of gravitational waves. Chapter 9 explains the main features of the Schwarzschild black hole, and applies General Relativity to Cosmology. The isotropic and homogeneous cosmological models are displayed within the context of the recent progress of the observational cosmology. The chapter ends with an updated account of the set of experimental results confirming General Relativity. Some special topics, not essential for a first reading of the book but meaningful for advanced students, are contained in the Appendix. In short, the final chapters are adequate for an introductory course on General Relativity and Cosmology. I wish to thank the Instituto de Astronomía y Física del Espacio (CONICET -UBA), where most of the lectures took place, for its hospitality. IAFE is the institute where a significant number of relativistic physicists from Buenos Aires got our start under the wise guidance of Mario Castagnino. I am indebted to Gerardo Milesi, Daniel Sforza, Claudio Simeone, and Marc Thibeault, who have contributed in several ways to the realization of this book: by adding biblio- graphic cites, correcting proofs, or enriching the contents with their comments. A special gratitude for my wife Mónica Landau—who has also collaborated in preparing this work—and my son Damián and my daughter Sofía, for their encouragement and patience during this long period of writing. Rafael Ferraro Buenos Aires, January 2005 Preface to the English Edition This English edition is basically the translation of the first Spanish edition published last year. A few changes and additions were made, and errors were corrected. I wish to express my gratitude to Luis Landau and Claudio Simeone for their cooperation in the realization of this edition. I am especially grateful to Alicia Semino for her thorough supervision of the English version of the manuscript. Rafael Ferraro Buenos Aires, October 2006 ix Contents List of Complements ................................................. xv List of Biographies...................................................xvii 1. Space and Time Before Einstein .................................. 1 1.1. Absolute Space and Time ............................................................