8083.9789814340489-tp.indd 1 8/29/11 4:59 PM This page intentionally left blank Bernard H Lavenda Universita’ degli Studi di Camerino, Italy World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI 8083.9789814340489-tp.indd 2 8/29/11 4:59 PM 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. A NEW PERSPECTIVE ON RELATIVITY An Odyssey in Non-Euclidean Geometries Copyright © 2012 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-13 978-981-4340-48-9 ISBN-10 981-4340-48-0 Typeset by Stallion Press Email: [email protected] Printed in Singapore. YeeSern - A New Perspective on Relativity.pmd1 10/14/2011, 9:08 AM Aug. 26, 2011 11:17 SPI-B1197 A New Perspective on Relativity b1197-fm In memory of Franco Fraschetti (1924–2009) v Aug. 26, 2011 11:17 SPI-B1197 A New Perspective on Relativity b1197-fm This page intentionally left blank Aug. 26, 2011 11:17 SPI-B1197 A New Perspective on Relativity b1197-fm Preface Electrodynamics was the next oasis after thermodynamics which saw a confluence of physicists and mathematicians, many of whom had been protagonists in thermodynamics. Just as thermodynamics had an offspring, quantum theory,so too did electrodynamics, namely the theory of relativity. While a single name can be attached to the origins of thermodynamics, Sadi Carnot, and that of its offspring, Max Planck, no such simplicity exists in electrodynamics and relativity. Relativity is as much about physics as it is about the human beings, and their failings, that made it. Every physics student will have heard of Maxwell’s equations but will he have also heard of Weber’s force? The stu- dent may have heard of Weber and Gauss for the units named after them, but not about their championing of Ampère’s law which threatened the supremacy of Newton’s inverse square law. The names of Helmholtz, Clau- sius and Boltzmann may be familiar from thermodynamics and statistical thermodynamics but much less known for their theories of electromag- netism. Every student of mathematics will have heard the names of Gauss and Riemann, but will he also know of their fundamental contributions to electromagnetism? Who were Abraham, Heaviside, Larmor, Liénard, Lorenz, Ritz, Schwarzschild, and Voigt? Why have their names been struck from the annals of electromagnetism? We are familiar with the priority disputes between Kelvin and Clausius in thermodynamics, but not with those in electromagnetism and relativity. A student of physics may have heard the name of Lorentz, because of his law of force and transformation, but not at the same level of Einstein. And Poincaré is known for just about everything else than his principle of relativity. The history of electromagnetism and relativity has been rewritten and in a very unflattering way. vii Aug. 26, 2011 11:17 SPI-B1197 A New Perspective on Relativity b1197-fm viii A New Perspective on Relativity By the modern historical account of electromagnetism and relativity, there were winners and losers. Maxwell is said to have triumphed over Weber and Gauss, in formulating a field theory of electromagnetism, and over Lorenz and Riemann in the formulation of his displacement current, Einstein’s absolute speed of light prevailed over Ritz’s ballistic theory of emission, Lorentz’s supremacy over Abraham and Bucherer in devising a model of the electron whose expressions for the variation of mass, momen- tum, and energy with velocity were later to be adopted in toto by relativity as a model for all matter, whether charged or not, and Einstein’s seniority in stating the principles of relativity though they were previously enunciated by Poincaré. Why were the experimentalists, Ives and Essen, so vehemently opposed to relativity? Ives viewed his verification of the second-order Doppler shift as a clear demonstration that a moving clock runs slow by the same factor that was predicted by Larmor and Lorentz, and not as a vindication of time dilatation in special relativity. Essen, who built the first cesium clock, queried what happens to the lost ticks when more ticks are transmitted than are received, independent of whether two clocks are approaching or receding from one another? Essen went so far as to query relativity as a “joke or swindle?” Most if not all monographs on relativity do not touch on these ques- tions. Not so with O’Rahilly’s Electromagnetics written in 1938. Not everyone will agree with his dispraise of Maxwell’s displacement current, or his over appraisal of Ritz, but much of what he says could not be truer today: There is far more authoritarianism in science that physicists are aware or at least publicly acknowledge. Anybody with a scientific reputation would today hesitate to criticize Einstein, except by way of outdoing him in cosmological speculations. Essen expressed similar views Students are told that the theory (relativity) must be accepted although they cannot be expected to understand it...The theory is so rigidly held that young scientists who have any regard for their careers dare not openly express their doubts. Whether there is any truth in the allegations I will leave to the reader. But what I plan to do is to present relativity from a ‘new’ point of view that treats, known and unknown, relativistic phenomena from different per- spectives. I put ‘new’ in quotation marks because the approach is really not new, but was suggested by Kaluza and Variˇcak over a century ago. What Aug. 26, 2011 11:17 SPI-B1197 A New Perspective on Relativity b1197-fm Preface ix is ‘new,’ I believe, is the wealth of physical phenomena that can be drawn from the non-Euclidean geometrical perspective. This monograph is nei- ther intended an historical account of relativity nor an essay in constructive criticism of it. Arecurring theme is that motion causes deformity and this can, under certain circumstances, catapult us into non-Euclidean spaces. It was also an exciting exercise to see where non-Euclidean geometries could be found, but were not appreciated as such. There are at least two eye-catching rela- tions: The product of two longitudinal Doppler shifts is the square root of the cross-ratio, and whose logarithm is hyperbolic distance, and the Beltrami metric in polar coordinates is the exact expression for the metric for the uniformly rotating disc. Gravitational phenomena rather than being a manifestation of warped space-time can be accounted for by a varying index of refraction in an inhomogeneous medium that modifies Fermat’s principle of least time. The reader will find old and new things alike — but the ‘old’ with a new interpretation. I don’t expect that everything is true to 100 percent, some things will have to be changed, modified or clarified, but, I do believe that this is a very fruitful approach that has led to the questioning of many fundamental aspects of relativity. According to Riemann, physics is the search for a geometric mani- fold upon which physical processes occur. The line element of constant curvature, 1 dx2 , (R) + 1 2 1 4 α x appearing in his Habilitation Dissertation, when written in polar coordinates is precisely the metric for a uniformly rotating disc with constant negative curvature, α<0. When charge is added, it becomes the Liénard expression for the rate of energy loss due to radiation. The role of the longitudinal Doppler shift means that space and time do not appear separately but only in a ratio, as a homogeneous coor- dinate. It is the difference in longitudinal Doppler shifts that is responsi- ble for the slowing down of clocks in relative motion. Einstein elevated c, the velocity of light in vacuo, to a universal constant. The fact that c is a constant, even to observers in relative motion, is tantamount to making it a unit of measurement — one which is necessary for the existence of a Aug. 26, 2011 11:17 SPI-B1197 A New Perspective on Relativity b1197-fm x A New Perspective on Relativity non-Euclidean geometry. So raising c to a universal constant, as Essen has pointed out, meant that the definition of unit length or time, or both, had to be abandoned. Relativity will thus unfold in a hyperbolic space of velocities that is entirely consonant with the relativistic addition of velocities. Following the historical route spreads the honors of discovery of rel- ativity more evenly. Poincaré had arrived at the postulates of relativity at least five years before Einstein, but “because he did not fully appreciate the status of both postulates” is no argument to deny him credit. To deny Poincaré his primary role in developing the theory of relativ- ity because he held onto the aether concept is to deny Carnot the credit for discovering his principle because he still believed in caloric theory.It would never have passed my mind to say that Boltzmann’s principle is incom- plete because it deals with only part of a probability distribution, being a very large number instead of a proper fraction, whereas I have shown that the entropy is the potential of law of error for which the most probable value is the average value of the measurements, that I have detracted any credit from Boltzmann.