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The Project Gutenberg EBook of Space--Time--Matter, by Hermann Weyl This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org Title: Space--Time--Matter Author: Hermann Weyl Translator: Henry L. Brose Release Date: June 21, 2013 [EBook #43006] Language: English Character set encoding: ISO-8859-1 *** START OF THIS PROJECT GUTENBERG EBOOK SPACE--TIME--MATTER *** Produced by Andrew D. Hwang, using scanned images and OCR text generously provided by the University of Toronto Gerstein Library through the Internet Archive. transcriber’s note The camera-quality files for this public-domain ebook may be downloaded gratis at www.gutenberg.org/ebooks/43006. This ebook was produced using scanned images and OCR text generously provided by the University of Toronto Gerstein Library through the Internet Archive. Typographical corrections, uniformization of punctuation, and minor presentational changes have been effected without comment. This PDF file is optimized for screen viewing, but may be recompiled for printing. Please consult the preamble of the LATEX source file for instructions and other particulars. SPACE—TIME—MATTER BY HERMANN WEYL TRANSLATED FROM THE GERMAN BY HENRY L. BROSE WITH FIFTEEN DIAGRAMS METHUEN & CO. LTD. 36 ESSEX STREET W.C. LONDON First Published in 1922 FROM THE AUTHOR’S PREFACE TO THE FIRST EDITION Einstein’s Theory of Relativity has advanced our ideas of the structure of the cosmos a step further. It is as if a wall which sep- arated us from Truth has collapsed. Wider expanses and greater depths are now exposed to the searching eye of knowledge, regions of which we had not even a presentiment. It has brought us much nearer to grasping the plan that underlies all physical happening. Although very recently a whole series of more or less popular introductions into the general theory of relativity has appeared, nevertheless a systematic presentation was lacking. I therefore considered it appropriate to publish the following lectures which I gave in the Summer Term of 1917 at the Eidgen. Technische Hochschule in Zürich. At the same time it was my wish to present this great subject as an illustration of the intermingling of philo- sophical, mathematical, and physical thought, a study which is dear to my heart. This could be done only by building up the theory systematically from the foundations, and by restricting at- tention throughout to the principles. But I have not been able to satisfy these self-imposed requirements: the mathematician pre- dominates at the expense of the philosopher. The theoretical equipment demanded of the reader at the out- set is a minimum. Not only is the special theory of relativity dealt with exhaustively, but even Maxwell’s theory and analytical ge- ometry are developed in their main essentials. This was a part of the whole scheme. The setting up of the Tensor Calculus—by means of which, alone, it is possible to express adequately the physical knowledge under discussion—occupies a relatively large amount of space. It is therefore hoped that the book will be found suit able for making physicists better acquainted with this math- ematical instrument, and also that it will serve as a text-book for students and win their sympathy for the new ideas. HERMANN WEYL Ribbitz in Mecklenburg Easter, 1918 PREFACE TO THE THIRD EDITION Although this book offers fruits of knowledge in a refrac- tory shell, yet communications that have reached me have shown that to some it has been a source of comfort in troublous times. To gaze up from the ruins of the oppressive present towards the stars is to recognise the indestructible world of laws, to strengthen faith in reason, to realise the “harmonia mundi” that transfuses all phenomena, and that never has been, nor will be, disturbed. My endeavour in this third edition has been to attune this har- mony more perfectly. Whereas the second edition was a reprint of the first, I have now undertaken a thorough revision which af- fects Chapters II and IV above all. The discovery by Levi-Civita, in 1917, of the conception of infinitesimal parallel displacements suggested a renewed examination of the mathematical foundation of Riemann’s geometry. The development of pure infinitesimal geometry in Chapter II, in which every step follows quite natu- rally, clearly, and necessarily, from the preceding one, is, I believe, the final result of this investigation as far as the essentials are concerned. Several shortcomings that were present in my first account in the Mathematische Zeitschrift (Bd. 2, 1918) have now been eliminated. Chapter IV, which is in the main devoted to Ein- stein’s Theory of Gravitation has, in consideration of the various important works that have appeared in the meanwhile, in partic- ular those that refer to the Principle of Energy-Momentum, been subjected to a very considerable revision. Furthermore, a new theory by the author has been added, which draws the physical inferences consequent on the extension of the foundations of geom- etry beyond Riemann, as shown in Chapter II, and represents an attempt to derive from world-geometry not only gravitational but also electromagnetic phenomena. Even if this theory is still only in its infant stage, I feel convinced that it contains no less truth than Einstein’s Theory of Gravitation—whether this amount of truth is unlimited or, what is more probable, is bounded by the Quantum Theory. I wish to thank Mr. Weinstein for his help in correcting the proof-sheets. HERMANN WEYL Acla Pozzoli, near Samaden August, 1919 PREFACE TO THE FOURTH EDITION In this edition the book has on the whole preserved its general form, but there are a number of small changes and additions, the most important of which are: (1) A paragraph added to Chapter II in which the problem of space is formulated in conformity with the view of the Theory of Groups; we endeavour to arrive at an un- derstanding of the inner necessity and uniqueness of Pythagorean space metrics based on a quadratic differential form. (2) We show that the reason that Einstein arrives necessarily at uniquely de- termined gravitational equations is that the scalar of curvature is the only invariant having a certain character in Riemann’s space. (3) In Chapter IV the more recent experimental researches dealing with the general theory of relativity are taken into consideration, particularly the deflection of rays of light by the gravitational field of the sun, as was shown during the solar eclipse of 29th May, 1919, the results of which aroused great interest in the theory on all sides. (4) With Mie’s view of matter there is contrasted an- other (vide particularly § 32 and § 36), according to which matter is a limiting singularity of the field, but charges and masses are force-fluxes in the field. This entails a new and more cautious attitude towards the whole problem of matter. Thanks are due to various known and unknown readers for pointing out desirable modifications, and to Professor Nielsen (at Breslau) for kindly reading the proof-sheets. HERMANN WEYL Zürich, November, 1920 TRANSLATOR’S NOTE In this rendering of Professor Weyl’s book into English, pains have been taken to adhere as closely as possible to the original, not only as regards the general text, but also in the choice of English equivalents for technical expressions. For example, the word affine has been retained. It is used by Möbius in his Der Barycentrische Calcul, in which he quotes a Latin definition of the term as given by Euler. Veblen and Young have used the word in their Projective Geometry, so that it is not quite unfamiliar to En- glish mathematicians. Abbildung, which signifies representation, is generally rendered equally well by transformation, inasmuch as it denotes a copy of certain elements of one space mapped out on, or expressed in terms of, another space. In some cases the Ger- man word is added in parenthesis for the sake of those who wish to pursue the subject further in original papers. It is hoped that the appearance of this English edition will lead to further efforts towards extending Einstein’s ideas so as to embrace all physical knowledge. Much has been achieved, yet much remains to be done. The brilliant speculations of the latter chapters of this book show how vast is the field that has been opened up by Einstein’s genius. The work of translation has been a great pleasure, and I wish to acknowledge here the courtesy with which suggestions concern- ing the type and the symbols have been received and followed by Messrs. Methuen & Co. Ltd. Acting on the advice of interested mathematicians and physicists I have used Clarendon type for the vector notation. My warm thanks are due to Professor G. H. Hardy of New College and Mr. T. W. Chaundy, M.A., of Christ Church, for valuable suggestions and help in looking through the proofs. Great care has been taken to render the mathematical text as perfect as possible. HENRY L. BROSE Christ Church, Oxford December, 1921 CONTENTS PAGE Introduction................................................. 1 CHAPTER I Euclidean Space. Its Mathematical Formulation and its Rôle in Physics § 1. Deduction of the Elementary Conceptions of Space from that of Equality . 15 § 2. The Foundations of Affine Geometry . 23 § 3. The Conception of n-dimensional Geometry. Linear Algebra. Quadratic Forms . 32 § 4. The Foundations of Metrical Geometry . 39 § 5. Tensors . 49 § 6. Tensor Algebra. Examples . 63 § 7. Symmetrical Properties of Tensors . 79 § 8. Tensor Analysis.
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