Table Of ContentFOUNDATIONS AND PHILOSOPHY OF SCIENCE AND
TECHNOLOGY SERIES
General Editor. MARIO BUNGE
McGill University, Montreal, Canada
Other Titles in the Series
AGASSI, J.
The Philosophy of Technology
ALCOCK, J. E.
Parapsychology—Science or Magic?
ANGEL, R.
Relativity: The Theory and its Philosophy
BUNGE, M.
The Mind-Body Problem
GIEDYMIN, J.
Science and Convention
HATCHER, W.
The Logical Foundations of Mathematics
SIMPSON, G.
Why and How: Some Problems and Methods in Historical Biology
WILDER, R.
Mathematics as a Cultural System
Pergamon Journals of Related Interest
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Relativity
and Geometry
by
ROBERTO TORRETTI
Department of Philosophy, University of Puerto Rico
PERGAMON PRESS
OXFORD NEW YORK · TORONTO SYDNEY PARIS FRANKFURT
U.K. Pergamon Press Ltd., Headington Hill Hall,
Oxford OX3 OBW, England
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75240 Pans, Cedex 05, France
FEDERAL REPUBLIC Pergamon Press GmbH, Hammerweg 6,
OF G E RM AN Y D-6242 Kronberg-Taunus, Federal Republic of Germany
Copyright c 1983 Roberto Torretti
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, electrostatic,
magnetic tape, mechanical, photocopying, recording
or otherwise, without permission in writing from the
publishers
First edition 1983
Reprinted 1984 (twice)
Library of Congress Cataloging in Publication Data
Torretti, Roberto, 1930-
Relativity and geometry.
(Foundations and philosophy of science and
technology series)
Includes bibliographical references and index.
1. Relativity (Physics) 2. Geometry, Differential.
3. Geometry—Philosophy. I. Title. II. Series:
Foundations & philosophy of science & technology.
QC173.55.T67 1983 530.l'l 82 9826
British Library Cataloguing in Publication Data
Torretti, Roberto
Relativity and geometry.—(Foundations and
philosophy of science and technology series)
1. Relativity 2. Geometry
I. Title II. Series
530. ΓI QC173.55
ISBN 0-08-026773-4
Printed in Great Britain by A. Wheaton & Co. Ltd, Exeter
For Christian
Preface
THE AIM and contents of the book are briefly described in the Introduction.
My debt to those who have written on the same subject before me is recorded
in the footnotes and in the list of references on pp. 351-379. Here I wish to
acknowledge my obligation and to convey my warmest thanks to those who
knowingly and willingly have aided me in my work.
Mario Bunge encouraged me to write the book for this series. I also owe him
much of my present understanding of fundamental physics as natural
philosophy, in the spirit of Aristotle, Newton and Einstein, and in opposition
to my earlier instrumentalist leanings.
Carla Cordua stood by me during the long and not always easy time of
writing, and gracefully put up with my absentmindedness. She heard much of
the book while still half-baked, helping me with her fine philosophical sense
and her keen awareness of style to shape my thoughts and to find the words to
express them.
John Stachel read the first draft of Chapter 5 and contributed with his vast
knowledge and his incisive intelligence to dispel some of my confusions and to
correct my errors. He wrote the paragraphs at the end of Section 5.8 (pp. 181
fi.) on the relation between gravitational radiation and the equations of
motion in General Relativity, an important subject of current research I did
not feel competent to deal with.
Adolf Grünbaum, whose "Geometry, Chronometry and Empiricism"
awakened my interest in the philosophical problems of relativity and
geometry, has kindly authorized me to quote at length from his writings.
David Malament corrected a serious error in my treatment of causal spaces.
Roger Angel, Mario Bunge, Alberto Coffa, Clark Glymour, Adolf
Grünbaum, Peter Havas, Bernulf Kanitscheider, David Malament, Lewis
Pyenson, John Stachel and Elie Zahar have sent me copies of some of their
latest writings. John Stachel, Kenneth Schaffner and my dear teacher Felix
Schwartzmann have called my attention to some papers I would otherwise
have overlooked. My son Christian, to whom the book is dedicated, has often
verified references and procured me materials to which I did not have direct
access.
The Estate of Albert Einstein has kindly authorized me to reproduce the
quotations from Einstein's correspondence. Part of them I obtained in the
Manuscript Room of the Firestone Library in Princeton University with the
expert and polite assistance of its staff. During my short stay in Princeton I
vii
VU! PREFACE
enjoyed the pleasant hospitality of the Institute for Advanced Study.
I began to write in August 1979 and finished today. In the second half of
1980 the University of Puerto Rico freed me from my teaching and
administrative duties and the John Simon Guggenheim Memorial
Foundation granted me a fellowship to pursue my work. I am deeply grateful
to both institutions for their renewed trust and their generous support.
Rio PIEDRAS, August 15th, 1981.
Naturwissenschaft ist der Versuch die Natur durch genaue Begriffe
aufzufassen.
BERNHARD RIEMANN
Nach unserer bisherigen Erfahrung sind wir nämlich zu dem Vertrauen
berechtigt, dass die Natur die Realisierung des mathematisch denkbar
Einfachsten ist. Durch rein mathematische Konstruktion vermögen
wir nach meiner Ueberzeugung diejenigen Begriffe und diejenige
gesetzliche Verknüpfung zwischen ihnen zu finden, welche den
Schlüssel für das Verstehen der Naturerscheinungen liefern.
ALBERT EINSTEIN *
Introduction
ι
GEOMETRY grew in ancient Greece as the science of plane and solid figures.
Purists would have had its scope restricted to figures that can be constructed
with ruler and compass, but its subject-matter somehow led by itself to the
study of curves such as the quadratrix (Hippias, 5th century B.C.), the spiral
(Archimedes, 287-212 B.C.), the conchoid (Nicomedes, ca. 200 B.C.), the
cissoid (Diocles, ca. 100 B.C.), which do not fit into that description. The
construction of a figure brings out points, which one then naturally regards as
pre-existing—at least "potentially"—in a continuous medium that provides, so
to speak, the material from which the figures are made. Understandably,
geometry came to be conceived of as the study of this medium, the science of
points and their relations in space. This idea is in a sense already implicit in the
Greek problem of loci—i.e. the search for point-sets satisfying some pre-
scribed condition. It is fully and openly at work in Descartes' Géométrie (1637).
When space itself, that is the repository of all points required for carrying out
the admissible constructions of geometry, was identified with matter
(Descartes), or with the locus of the divine presence in which matter is placed
(Henry More), the stage was set for the foundation of natural philosophy upon
geometrical principles (Newton). Inspired to a considerable extent by
Newton's success, Kant developed his view of geometry as a paradigm of our
a priori knowledge of nature, the surest proof that the order of nature is an
outgrowth of human reason.
For all its alluring features, Kant's philosophy of geometry could not in the
long run stand its ground before the onrush of the many geometries that
flourished in the 19th century. Bold yet entirely plausible variations of the
established principles and methods of geometrical thinking gave rise to a
wondrous crop of systems, that furnished either generalizations and exten-
sions of, or alternatives to, the geometry of Euclid and Descartes. The almost
indecent fertility of reason definitely disqualified it from prescribing the
geometrical structure of nature. The choice of a physical geometry, among the
many possibilities offered by mathematics, was either a matter of fact, to be
resolved by experimental reasoning (Gauss, Lobachevsky), or a mere matter of
agreement (Poincaré).
Two widely different proposals were made, in the third quarter of the
century, for ordering and unifying the new disciplines of geometry; namely,
1
2 RELATIVITY AND GEOMETRY
Riemanrfs theory of manifolds, and Klein's Erlangen programme. 1 Riemann
sought to classify physical space within the vast genus of structured sets or
"manifolds" of which it is an instance. While countenancing the possibility that
it might ultimately be discrete, he judged that, at any rate macroscopically, it
could be regarded as a three-dimensional continuum (i.e., in current parlance,
as a topological space patchwise homeomorphic to R 3). Lines, i.e. one-
dimensional continua embedded in physical space, had a definite length
independent of the manner of their embedding. The practical success of
Euclidean geometry suggested, moreover, that the infinitesimal element of
length was equal, at each point of space, to the positive square root of a
quadratic form in the coordinate differentials, whose coefficients, however,
would in all likelihood—for every choice of a coordinate system—vary
continuously with time and place, in utter contrast with the Euclidean case.
Riemann's theory of metric manifolds was perfected by Christoffel, Schur,
Ricci-Curbastro, etc. and would probably have found an application to
classical mechanics if Heinrich Hertz had been granted the time to further
develop his ideas on the subject. But it was not taken seriously as the right
approach to physical geometry until Albert Einstein based on it his theory of
gravitation or, as he preferred to call it, the General Theory of Relativity.
Klein defined a geometry within the purview of his scheme by the following
task: Let there be given a set and a group acting on it; to investigate the
properties and relations on the set which are not altered by the group
transformations.2 A geometry is thus characterized as the theory of invariants
of a definite group of transformations of a given abstract set. 3 The known
geometries can then be examined with a view to ascertaining the transform-
ation group under which the properties studied by it are invariant. Since the
diverse groups are partially ordered by the relation of inclusion, the geometries
associated with each of them form a hierarchical system. Klein's elegant and
simple approach won prompt acceptance. It encouraged some of the deepest
and most fruitful mathematical research of the late 19th century (Lie on Lie
groups; Poincaré on algebraic topology). It stimulated the advent of the
conventionalist philosophy of geometry. 4 Its most surprising and perhaps in
the long run historically most decisive application was the discovery by
Hermann Minkowski that the modified theory of space and time on which
Albert Einstein (1905c/) had founded his relati vistic electrodynamics of moving
bodies was none other than the theory of invariants of a definite group of
linear transformations of R 4, namely, the Lorentz group. 5 It was thus shown
that the new non-Newtonian physics—known to us as the Special Theory of
Relativity—that was being then systematically developed by Einstein, Planck,
Minkowski, Lewis and Tolman, Born, Laue, etc., rested on a new, non-
Euclidean, geometry, which incorporated time and space into a unified
"chronogeometric" structure. Minkowski's discovery not only furnished a
most valuable standpoint for the understanding and fruitful development of
INTRODUCTION 3
Special Relativity, but was also the key to its subsequent reinterpretation as the
local—tangential—approximation to General Relativity. The latter, of course,
was conceived from the very beginning as a dynamical theory of spacetime
geometry—of a geometry, indeed, of which the Minkowski geometry is a
unique and rather implausible special case.
The main purpose of this book is to elucidate the motivation and
significance of the changes in physical geometry brought about by Einstein, in
both the first and the second phase of Relativity. However, since the geometry
is, in either phase, inseparable from the physics, what the book in fact has to
offer is a "historico-critical" exposition of the elements of the Special and the
General Theory of Relativity. But the emphasis throughout the book is on
geometrical ideas, and, although I submit that the standpoint of geometry is
particularly congenial to the subject, I also believe that there is plenty of room
left for other comparable studies that would regard it from a different, more
typically "physical" perspective.6
2
The book consists of seven chapters and a mathematical appendix. The first
two chapters review some of the historical background to Relativity. Chapter 1
deals with the principles of Newtonian mechanics—space and time, force and
mass—as they apply, in particular, to the foundation of Newtonian kine-
matics. Chapter 2 summarily sketches some of the relevant theories and results
of pre-Relativity optics and electrodynamics. 7 The next two chapters refer to
Special Relativity. Chapter 3 is mainly devoted to Einstein's first Relativity
paper of 1905. It carefully analyses Einstein's definition of time on an inertial
frame—which, as I will show, can be seen as a needful improvement of the
Neumann-Lange definition of an inertial time scale—and his first derivation
of the Lorentz transformation. It also comments on some implications of the
Lorentz transformation, such as the relativistic effects usually described as
"length contraction" and "time dilation"; on the alternative approach to the
Lorentz transformation initiated by W. von Ignatowsky (1910), and on the
relationship between Einstein's Special Theory of Relativity and the con-
temporary, predictively equivalent theory of Lorentz (1904) and Poincaré
(1905, 1906). Chapter 4 presents the Minkowskian formulation of Special
Relativity with a view to preparing the reader for the subsequent study of
General Relativity. Thus, Sections 4.2 and 4.3 (together with Sections A and Β
of the Appendix) explain the concept of a Riemannian Η-manifold and of a
geometric object on an H-manifold, and Section 4.5 introduces von Laue's
stress-energy-momentum tensor. Section 4.6 sketches the theory of causal
spaces originating with A. A. Robb (1914) and perfected in our generation by
E. Kronheimer and R. Penrose (1967). Chapter 5 deals with Einstein's search
for General Relativity from 1907 to 1915. As I am not altogether satisfied with
