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Geometry, Relativity, and the Fourth Dimension

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Geometry, Relativity, and the Fourth Dimension %* GEOMETRY, RELATIVITY AND THE FOURTH I I DIMENSION •^ ^^f!^ 1 DOVER BOOKS ON RELATIVITY AND RELATED AREAS Methods of Quantum Field Theory in Statistical Physics, A.A. Abriko- sov, et al. (63228-8) $6.00 The Evolution of Scientific Thought from Newton to Einstein, A. d'Abro. (20002-7) $6.00 The Rise of the New Physics, A. d'Abro. (20003-5, 20004-3) Two-volume set $12.00 Introduction to the Theory of Relativity, Peter G. Bergmann. (63282-2) $4.50 Relativity and Common Sense, Hermann Bondi. (24021-5) $3.00 Einstein's Theory of Relativity, Max Bom. (60769-0) $4.50 The Restless Universe, Max Bom. (20412-X) $6.00 Causality and Modern Science, Mario Bunge. (23728-1) $6.95 Investigations on the Theory of the Brownian Movement, Albert Ein- stein. (60304-0) $2.75 The Principle of Relativity, Albert Einstein, Hendrik Lorentz, H. Min- kowski, and Hermann Weyl. (60081-5) $3.50 Differential Geometry, Heinrich W. Guggenheimer. (63433-7) $6.00 The Physical Principles of the Quantum Theory, Wemer Heisenberg. (60 1 1 3-7) $3.50 Atomic Spectra and Atomic Structure, Gerhard Herzberg. (60115-3) $4.00 Speculations on the Fourth Dimension, C. H. Hinton. (23916-0) $4.00 The Domain of Natural Science, E.W. Hobson. (21966-6) $3.50 The Strange Story of the Quantum, Banesh Hoffmaim. (20518-5) $4.25 The Nuclear Properties of the Heavy Elements, Earl K. Hyde, et al. (62805-1, 62806-X, 62807-8) Three-volume set, Clothbound $45.00 Crucibles: The Story of Chemistry, Bernard Jaffe. (23342-1) $5.50 The Absolute Differential Calculus, Tullio Levi-Civita. (63401-9) $7.00 Electricity and Magnetism, James Clerk Maxwell. (60636-8, 60637-6) Two-volume set $13.(X) Fundamental Formulas of Physics, Donald H. Menzel. (60595-7, 60596-5) Two-volume set $14.00 Mathematical Physics, Donald H. Menzel. (6(X)56-4) $6.(X) The Theory of the Properties of Metals and Alloys, N.F. Mott and H. Jones. (60456-X) $6.00 Theory of Relativity, Wolfgang Pauli. (64152-X) $4.00 From Copernicus to Einstein, Hans Reichenbach. (23940-3) $2.25 The Philosophy of Space and Time, Hans Reichenbach. (60443-8) $4.75 Geometry, Relativity and the Fourth Dimension, Rudolf Rucker. (23400-2) $2.75 Selected Papers on Quantum Electrodynamics, Jufian Schwinger (ed.). (60444-6) $7.50 {continued on back flap) fe. 5" _ GEOMETRY, RELATIVITY and the FOURTH DIMENSION GEOMETRY, RELATIVITY and the FOURTH DIMENSION by Rudolf v.B. Rucker Department of Mathematics, State University College of Arts and Science, Geneseo, N.Y. Dover Publications, Inc., New York Copyright © 1977 by Rudolf v.B. Rucker. All rights reserved under Pan American and Interna- tional Copyright Conventions. Published in Canada by General Publishing Com- pany, Ltd., 30 Lesmill Road, Don Mills, Toronto, Ontario. Pubhshed in the United Kingdom by Constable and Company, Ltd., 10 Orange Street, London WC2H 7EG. Geometry, Relativity and the Fourth Dimension is a new work, first pubhshed by Dover Pubhcations, Inc., in 1977. International Standard Book Number: 0-486-23400-2 Library of Congress Catalog Card Number: 76-22240 Manufactured in the United States of America Dover Pubhcations, Inc. 180 Varick Street New York, N. Y. 10014 PREFACE This book is about the fourth dimension and the structure of our universe. My goal has been to present an intuitive picture of the curved space-time we call home. There are a number of excellent introductions to the separate topics treated here, but there has been no prior weaving of them into a sustained visual account. I looked for a book like this for many years—and finding none, I wrote it. Geometry, Relativity and the Fourth Dimension is written in the hope that any interested person can enjoy it. I would only advise the casual reader to be willing to skim through those few sections that may seem too purely mathematical. This book is, however, more than a standard popular exposition. There is a great deal of original material here, and even the experienced mathematician or physicist will find unexpected novelties. I am indebted to all of the authors whose work is described in the Bibhography, but most especially to Edwin Abbott, Arthur Eddington, Hans Reichenbach and John Wheeler. R. V. B. R. Geneseo, N.Y. January 31, 1976 CONTENTS 1 : The Fourth Dimension 1 2: Non-Eudidean Geometry 17 3: Curved Space 36 4: Time as a Higher Dimension 57 5: Special Relativity 68 6: Time Travel 92 7: The Shape of Space-Time 100 Conclusion ~ 117 Annotated Bibliography 119 GEOMETRY, RELATIVITY and the FOURTH DIMENSION 1 THE FOURTH DIMENSION We live in three-dimensional space. That is, motion in our space has three degrees of freedom—no fewer and no more. In other words, we have three mutually perpendicular types of motion (left/right, for- ward/backward, up/down), and any point in our space can be reached by combining the three possible types of motion (e.g., "Walk straight ahead about 200 paces to the river, then go right about 50 paces until you come to a big oak tree. Chmb about 40 feet up it. I'll be waiting for you there."). Normally it is difficult for us to perform up/down motions; space is more three-dimensional for a bird or a fish than it is for us. On the other hand, space is essentially one-di- mensional for a car driving down a two-lane road, essentially two-di- mensional for a snowmobile or a car driving around an empty parking lot. How could there be a fourth dimension, a direction perpendicu- lar to every direction that we can indicate in our three-dimensional space? In order to get a better understanding of what a "fourth dimension" might mean, consider the following sequence: We take a 0-D point (Figure 1; from now on, we'll abbreviate "« -dimensional" by "n-D"), move the point one unit to the right (this produces a 1-D line segment, Figure 2), move this segment one unit downward (this, with the lines connecting the old and new segments, produces a 2-D square. Figure 3) and move the square one unit forward out of the paper to produce a 3-D cube (Figure 4). Notice that we cannot actually draw a 3-D cube on this 2-D sheet of paper. We represent the third dimension by a line that is diagonal (rather than perpendicular) to the left/right and up/down dimensions. Now, we don't really know anything yet about the fourth 2 / Geometry, Relativity and the Fourth Dimension y // Fig. 1. Fig. 2. Fig. 3. / 7 /Fig. 4. dimension, but couldn't we try representing it by a direction on the paper that is perpendicular to the (diagonal) direction we used to represent the third dimension? If we do so, we can continue our sequence by moving the cube one unit in the direction of the fourth dimension, producing a 4-D hypercube (Figure 5). /\ /\ V KX / /X>A<X\ X \/ \/Fig. 5. This design for the hypercube is taken from a Uttle 1913 book, A Primer of Higher Space, by Claude Bragdon, an architect who incor- porated this and other 4-D designs into such structures as the Rochester Chamber of Commerce Building. It is also possible to consider a similar sequence of spheres of various dimensions. A sphere is given by its center and its radius; thus the sphere with center and radius 1 is the set of all points P such that the distance between and P is \. This definition is independent of the number of dimensions your space has. There is no The Fourth Dimension I 3 such thing as a 0-D sphere of radius 1, since a 0-D space has only one point. A 1-D sphere of radius 1 around consists of two points (Figure 6). -1 ' -I ii, W=i Fig. 6. A 2-D sphere of radius 1 can be represented by this figure in the jcy-plane (Figure 7). x^+y^=l x-^ + y^ + z^= 1 Fig. 7. Fig. 8. A 3-D sphere of radius 1 in the xyz coordinate system looks like Figure 8. Although, reasoning by analogy, a 4-D sphere (Jiypersphere) can be seen to be the set of quadruples {x,y, z, t) such that x^+y-^-\- z^ + /^ = 1 in the xyzt coordinate system, we cannot say that we have a very good mental image of the hypersphere. Interestingly, mathemati- cal analysis does not require an image, and we can actually use calculus to find out how much 4-D space is inside a hypersphere of a given radius r. The 1-D space inside a 1-D sphere of radius r is the length 2r. The 2-D space inside a 2-D sphere of radius r is the area irp-. The 3-D space inside a 3-D sphere of radius r is the volume 4/3 irr- The 4-D space inside a 4-D sphere of radius r is the hypervolume 1/2 ttV. One of the most effective methods for imagining the fourth dimension is the method of analogy. That is, in trying to imagine how 4-D objects might appear to us, it is a great help to consider the analogous efforts of a 2-D being to imagine how 3-D objects might appear to him. The 2-D being whose efforts we will consider is named A. Square (Figure 9) and he lives in Flatland. A. Square first appeared in the book Flatland, written by Edwin 4 / Geometry, Relativity and the Fourth Dimension Fig.
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