the frontiers collection the frontiers collection Series Editors: D Dragoman M Dragoman A.C Elitzur M.P Silverman J Tuszynski H.D Zeh The books in this collection are devoted to challenging and open problems at the forefront of modern physics and related disciplines, including philosophical debates In contrast to typical research monographs, however, they strive to present their topics in a manner accessible also to scientifically literate non-specialists wishing to gain insight into the deeper implications and fascinating questions involved Taken as a whole, the series reflects the need for a fundamental and interdisciplinary approach to modern science It is intended to encourage scientists in all areas to ponder over important and perhaps controversial issues beyond their own speciality Extending from quantum physics and relativity to entropy, time and consciousness – the Frontiers Collection will inspire readers to push back the frontiers of their own knowledge Information and Its Role in Nature By J.G Roederer Relativity and the Nature of Spacetime By V Petkov Quo Vadis Quantum Mechanics? Edited by A C Elitzur, S Dolev, N Kolenda Life – As a Matter of Fat The Emerging Science of Lipidomics By O.G Mouritsen Quantum–Classical Analogies By D Dragoman and M Dragoman Knowledge and the World Challenges Beyond the Science Wars Edited by M Carrier, J Roggenhofer, G Kăuppers, P Blanchard QuantumClassical Correspondence By A.O Bolivar Mind, Matter and Quantum Mechanics By H Stapp Quantum Mechanics and Gravity By M Sachs V Petkov RELATIVITY AND THE NATURE OF SPACETIME With 58 Figures 123 Dr Vesselin Petkov Concordia University Liberal Arts College de Maisonneuve Blvd West 1455 Montreal, Quebec H3G 1M8, Canada E-mail: vpetkov@alcor.concordia.ca Series Editors: Prof Daniela Dragoman University of Bucharest, Physics Faculty, Solid State Chair, PO Box MG-11, 76900 Bucharest, Romania email: danieladragoman@yahoo.com Prof Mircea Dragoman National Research and Development Institute in Microtechnology, PO Box 38-160, 023573 Bucharest, Romania email: mircead@imt.ro Prof Avshalom C Elitzur Bar-Ilan University, Unit of Interdisciplinary Studies, 52900 Ramat-Gan, Israel email: avshalom.elitzur@weizmann.ac.il Prof Mark P Silverman Department of Physics, Trinity College, Hartford, CT 06106, USA email: mark.silverman@trincoll.edu Prof Jack Tuszynski University of Alberta, Department of Physics, Edmonton, AB, T6G 2J1, Canada email: jtus@phys.ualberta.ca Prof H Dieter Zeh University of Heidelberg, Institute of Theoretical Physics, Philosophenweg 19, 69120 Heidelberg, Germany email: zeh@urz.uni-heidelberg.de Cover figure: Detail from ‘Venus Beauty and Anisotropic Geometric Diffusion’ by U Clarenz, U Diewald, and M Rumpf Courtesy of M Rumpf ISSN 1612-3018 ISBN-10 3-540-23889-1 Springer Berlin Heidelberg New York ISBN-13 978-3-540-23889-8 Springer Berlin Heidelberg New York Library of Congress Control Number: 2005924256 This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer Violations are liable to prosecution under the German Copyright Law Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2005 Printed in Germany The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Typesetting by Stephen Lyle using a Springer TEX macro package Final processing by LE-TEX Jelonek, Schmidt & Văockler GbR, Leipzig Cover design by KünkelLopka, Werbeagentur GmbH, Heidelberg Printed on acid-free paper SPIN: 10977378 57/3141/YL - To all who struggle to understand this strange world Preface The standard books on relativity not usually address the questions of the physical meaning of relativistic effects and the nature of spacetime This book deals specifically with such conceptual questions All kinematic consequences of special relativity are analyzed by explicitly asking whether the physical objects involved in these effects are threedimensional or four-dimensional; this is equivalent to asking whether those objects exist only at the present moment of their times, as our common sense suggests, or at all moments of their histories An answer to the question of the dimensionality of physical objects will resolve the issue of the nature of spacetime – whether spacetime is just a mathematical space (like a seven-dimensional color space, for instance) or represents a real four-dimensional world This book is intended for physicists, philosophers of science, philosophers, physics and philosophy students, and anyone who is interested in what special relativity is telling us about the world Acknowledgements I would like to express my gratitude to all who contributed to the appearance of this book So many people were involved in discussions on the issues covered here that it is virtually impossible to mention them all That is why I would merely like to thank them for their constructive comments and hope that our discussions have brought us a little closer to understanding this beautiful but strange world I feel I should start the short list of specific acknowledgements by thanking Springer and Dr Angela Lahee for starting the publication of The Frontiers Collection I think the appearance of such a series is more than timely since scientists have already started to lose sight of new developments in the various scientific fields I would also like to thank Stephen Lyle for his excellent technical editing of the manuscript I owe a lot to my teacher and friend Anastas Anastassov of Sofia University His excellent lectures on general relativity in the 1980s VIII Preface and our never-ending discussions prepared the ground for the ideas developed in this book My thanks also go to Prof Tzvetan Bonchev (at the time Dean of Sofia University’s Faculty of Physics and Chair of the Department of Atomic Physics) and Prof Ivanka Apostolova (at the time Chair of Sofia University’s Department of Philosophy) Their influence is difficult to estimate I am grateful to my colleagues from the Department of Philosophy of Science of the Institute for Philosophical Research at the Bulgarian Academy of Sciences with whom many of the topics in this book were discussed in the late 1980s Versions of the issues examined in the book have been covered in different classes I taught – in the philosophy of science classes at Sofia University in the 1980s and later in the physics and in the philosophy of science classes at Concordia University I am grateful to all students who participated in the class discussions I also benefited from valuable comments from colleagues and students at Concordia University, McGill University and the University of Montreal, who attended a series of lectures I gave at a weekly seminar on General Relativity in the Fall of 1994, held at Concordia University I would like to express my sincere thanks to all anonymous referees who made constructive recommendations and comments on different issues that are now included in this book Most of the results presented here were also reported at several international conferences and at two inter-university seminars in Montreal – on open questions in physics and on the history and philosophy of science I am truly grateful to the colleagues and students who took part in the discussions And last, I would like to express my deep gratitude to my wife Svetoslava and our son Vesselin (Jr) for their understanding, unconditional support, and encouragement The completion of this book would not have been possible without their endless love and faith in me Montreal, 12 October 2004 Vesselin Petkov Contents Introduction Part I From Galileo to Minkowski On the Impossibility of Detecting Uniform Motion 13 2.1 Aristotle’s View on Motion 14 2.2 Copernicus and Ptolemy’s Arguments Against the Earth’s Motion 16 2.3 Galileo’s Disproof of Aristotle’s View on Motion 17 2.4 Galileo’s Principle of Relativity 25 Exploring the Internal Logic of Galileo’s Principle of Relativity 29 3.1 On the Physical Meaning of Galileo’s Principle of Relativity 30 3.2 On the Two Postulates of Special Relativity 48 3.3 A Lesson from a Delayed Discovery 51 3.4 Summary 53 Relativity in Euclidean Space and in Spacetime 4.1 Spacetime 4.2 Derivation of the Lorentz Transformations 4.3 Four-Dimensional Distance and Three Kinds of Length 4.4 Y ‘Dilation’ in Euclidean Space and Time Dilation in Spacetime 4.5 Length Contraction in Euclidean Space and in Spacetime 4.6 The Twin Paradox in Euclidean Space and in Spacetime 4.7 Addition of Velocities 4.8 The Metric of Spacetime 55 56 70 78 84 91 98 105 106 References H Weyl: Philosophy of Mathematics and Natural Science (Princeton University Press, Princeton 1949) p 116 N Copernicus: On the Revolutions of the Heavenly Spheres In: Great Books of the Western World , Vol 15, ed by M.J Adler (Encyclopedia Britannica, Chicago 1993) J Kepler: Harmonies of the World , Book Five In: On the Shoulders of Giants: The Great Works of Physics and Astronomy, ed by S Hawking (Running Press, Philadelphia, London 2002) G Galileo: Dialogue Concerning the Two Chief World Systems – Ptolemaic and Copernican, 2nd edn (University of California Press, Berkeley 1967) J Barnes: Early Greek Philosophy, 2nd edn (Penguin Books, London 2001) Part II J Barnes: The Presocratic Philosophers (Routledge, London, New York 1982) Chap X Aristotle: Physics In: Great Books of the Western World, Vol 7, ed by M.J Adler (Encyclopedia Britannica, Chicago 1993) C Ptolemy: The Almagest In: Great Books of the Western World, Vol 15, ed by M.J Adler (Encyclopedia Britannica, Chicago 1993) pp 8–13 H Minkowski: Space and Time In: [10] pp 75–91 10 H.A Lorentz, A Einstein, H Minkowski, and H Weyl: The Principle of Relativity: A Collection of Original Memoirs on the Special and General Theory of Relativity (Dover, New York 1952) 11 A Einstein: On the Electrodynamics of Moving Bodies In: [10] pp 37– 65 12 A Sommerfeld: Notes on Minkowski’s paper: Space and Time In: [10] pp 92–96 13 W Pauli: Theory of Relativity (Dover, New York 1958) p 14 A.P French: Special Relativity (Norton, New York, London 1968) p 72 15 R Resnick, D Halliday: Basic Concepts in Relativity and Early Quantum Theory, 2nd edn (Macmillan, New York 1992) p 27 16 U.E Schră oder: Special Relativity (World Scientific, Singapore 1990) p 19 282 References 17 W.D McComb: Dynamics and Relativity (Oxford University Press, Oxford, New York 1999) p 187 18 C.H Hinton: What is the Fourth Dimension? (W.S Sonnenschein and Co., London 1884) 19 C.H Hinton: Speculations on the Fourth Dimension: Selected Writings (Dover, New York 1980) 20 R Le Poidevin: Travels in Four Dimensions: The Enigmas of Space and Time (Oxford University Press, Oxford, New York 2003) p 48 21 E.F Taylor, J.A Wheeler: Spacetime Physics: Introduction to Special Relativity, 2nd edn (Freeman, New York 1992) 22 W Rindler: Relativity (Oxford University Press, Oxford, New York 2001) 23 H Stein: Phil Sci 58, 147 (1991) 24 C.W Rietdijk: Phil Sci 33, 341 (1966) 25 H Putnam: J Phil 64, 240 (1967) 26 N Maxwell: Phil Sci 52, 23 (1985) 27 H Stein: J Phil 65, (1968) 28 D Dieks: Phil Sci 55, 456 (1988) 29 S McCall and E.J Lowe: Analysis 63, 114 (2003) 30 Y Balashov: On Stages, Worms, and Relativity In: [32] 31 C Callender: Phil Sci 67 (Proceedings), S587 (2000) 32 C Callender (Ed.): Time, Reality and Experience (Cambridge University Press, Cambridge 2002) 33 S Saunders: How Relativity Contradicts Presentism In: [32] 34 S 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d’Inverno: Introducing Einstein’s Relativity (Clarendon Press, Oxford 1992) 44 G.L Naber: The Geometry of Minkowski Spacetime (Springer, Berlin, Heidelberg, New York 1992) 45 P Kroes: Phil Sci 50, 159–163 (1983) 46 R Weingard: Brit J Phil Sci 23, 119 (1972) 47 V Petkov: Brit J Phil Sci 40, 69 (1989) 48 D Dieks: Space, Time and Coordinates in a Rotating World In: Relativity in Rotating Frames: Relativistic Physics in Rotating Reference Frames (Fundamental Theories of Physics, Vol 135), ed by G Rizzi, M.L Ruggiero (Kluwer, Dordrecht, Boston, London 2004) pp 29–42 49 Aristotle: Physics, Book IV, Chap 14 In: Great Books of the Western World , Vol 7, ed by M.J Adler (Encyclopedia Britannica, Chicago 1993) 50 Saint Augustine: The Confessions, Book XI In: Great Books of the Western World , Vol 16, ed by M.J Adler (Encyclopedia Britannica, Chicago 1993) 51 V Petkov: Weyl’s View on the Objective World In: Exact Sciences and Their Philosophical Foundations, ed by W Deppert, K Huebner, A Oberschelp, V Weidemann (P Lang, Frankfurm am Main Bern New York, Paris 1988) pp 519–524 52 A.H Anastassov: The Theory of Relativity and the Quantum Action (4-Atomism),DSc Thesis, Sofia University, Sofia (1984); The Theory of Relativity and the Quantum Action (Nautilus, Sofia 2003), in Bulgarian 53 A.H Anastassov: Annuaire de l’Universit´e de Sofia, St Kliment Ohridski, Facult´e de Physique, 81, 135 (1993) 54 Encyclopedia of Physics, 2nd edn, ed by R.G Lerner and G.L Trigg (VCH Publishers, New York 1991) p 721 55 J Trefil: The Nature of Science (Houghton Mifflin, Boston New York 2003) p 98 56 B.G Kuznetsov: Einstein and Dostoyevsky (Hutchinson, London 1972) 57 H Ohanian, R Ruffini: Gravitation and Spacetime, 2nd edn (W.W Norton, New York, London 1994) p 197 58 A Einstein, L Infeld: The Evolution of Physics (Simon and Schuster, New York 1966) p 221 59 P.A Tipler: Physics, Vol 3, 4th edn (Freeman, New York 1999) p 1272 60 R.L Reese: University Physics, Vol (Brooks/Cole, New York 2000) p 1191 61 R.A Serway: Physics, Vol 2, 4th edn (Saunders, Chicago 1996) p 1180 62 P.M Fishbane, S Gasiorowicz, S.T Thornton: Physics (Prentice Hall, New Jersey 1993) p 1192 63 G Rizzi, M.L Ruggiero (Eds.): Relativity in a Rotating Frame (Kluwer, Dordrecht 2003) 284 References 64 I.I Shapiro: Phys Rev Lett 13, 789 (1964) 65 I.I Shapiro: Phys Rev 141, 1219 (1966) 66 E.F Taylor, J.A Wheeler: Exploring Black Holes: Introduction to General Relativity (Addison Wesley Longman, San Francisco 2000) p E-1 67 A Harpaz, N Soker: Gen Rel Grav 30, 1217 (1998); see also physics/9910019 68 W Rindler: Am J Phys 36, 540 (1968) 69 A Einstein: Ann Phys 49 (1916) In: [10] pp 111–164 70 L Lerner: Am J Phys 65, 1194 (1997) 71 E Fermi: Nuovo Cimento 22, 176 (1921) 72 D.J Griffiths: Introduction to Electrodynamics, 2nd edn (Prentice Hall, London 1989) p 416 73 W.K.H Panofsky and M Phillips: Classical Electricity and Magnetism, 2nd edn (Addison-Wesley, Massachusetts, London 1962) p 342 74 R.P Feynman, R.B Leighton and M Sands: The Feynman Lectures on Physics, Vol (Addison-Wesley, New York 1964) p 21-10 75 M Schwartz: Principles of Electrodynamics (Dover, New York 1972) p 213 76 J.D Jackson, Classical Electrodynamics, 3rd edn (Wiley, New York 1999) p 664 77 F Rohrlich: Classical Charged Particles (Addison-Wesley, New York 1990) p 218 78 M von Laue: Relativită ats Theorie, 3rd edn., Vol (Frederick Vieweg und Sohn, Braunschweig 1919) 79 T Fulton, F Rohrlich: Ann Phys 9, 499 (1960) 80 F Rohrlich: Ann Phys 22, 169 (1963) 81 D.G Boulware: Ann Phys 124, 169 (1980) 82 B.S DeWitt and R.W Brehme: Ann Phys 9, 220 (1960) 83 W Rindler: Essential Relativity (Springer, Berlin, Heidelberg, New York 1997) p 244 84 J.L Synge: Relativity: The General Theory (Nord-Holland, Amsterdam 1960) p 109 85 J.J Thomson: Phil Mag 11, 229 (1881) 86 O Heaviside: The Electrician 14, 220 (1885) 87 G.F.C Searle: Phil Mag 44, 329 (1897) 88 M Abraham: The Classical Theory of Electricity and Magnetism, 2nd edn (Blackie, London 1950) 89 H.A Lorentz: Proceedings of the Academy of Sciences of Amsterdam 6, 809 (1904) 90 H.A Lorentz: Theory of Electrons, 2nd edn (Dover, New York 1952) 91 H Poincar´e: Compt Rend 140, 1504 (1905) 92 H Poincar´e: Rendiconti del Circolo Matematico di Palermo 21, 129 (1906) 93 E Fermi: Phys Zeits 23, 340 (1922) References 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 285 E Fermi: Rend Acc Lincei (5) 31, 184; 306 (1922) E Fermi: Nuovo Cimento 25, 159 (1923) H Mandel: Z Physik 39, 40 (1926) W Wilson: Proc Phys Soc 48, 736 (1936) M.H.L Pryce: Proc Roy Soc A 168, 389 (1938) B Kwal: J Phys Rad 10, 103 (1949) F Rohrlich: Am J Phys 28, 639 (1960) M Jammer: Concepts of Mass in Classical and Modern Physics (Dover, New York, 1997), Chap 11 See also [102] M Jammer: Concepts of Mass in Contemporary Physics and Philosophy (Princeton University Press, Princeton, 2000) p 34 J.W Butler: Am J Phys 37, 1258 (1969) E Mach: Science of Mechanics, 9th edn (Open Court, London 1933) P Pearle: Classical Electron Models In: Electromagnetism: Paths to Research, ed by D Teplitz (Plenum Press, New York 1982) pp 211– 295 D Bender et al: Phys Rev D30, 515 (1984) M.H MacGregor: The Enigmatic Electron (Kluwer, Dordrecht 1992) D Hestenes, A Weingartshofer (Eds.): The Electron: New Theory and Experiment (Kluwer, Dordrecht 1991) M Springford (Ed.): Electron: A Centenary Volume (Cambridge University Press, Cambridge 1997) B Podolsky, K.S Kunz: Fundamentals of Electrodynamics (Marcel Dekker, New York 1969) p 288 M.K Gaillard, P.D Grannis, and F.J Sciulli: Rev Mod Phys 71, No S96 (1999) A Einstein: Ann Phys 20, 627 (1906) D.L Livesey: Atomic and Nuclear Physics (Blaisdell, Massachusetts 1966) p 117 D J Griffiths and R E Owen: Am J Phys 51, 1120 (1983) Index 3D/4D dilemma, 123, 124, 142 4/3 factor, 255, 258, 274, 275 Abraham, M., 252 absolute motion, 49 does not exist, 50 absolute rest, 49 absolute space, 49 absolutism, 31 acceleration, absoluteness of, 189, 191 Anastassov, A., 162 area of applicability of scientific theory, 119 Aristarchus of Samos, 13, 16 Aristotelian doctrine of motion, 17, 24 Aristotle, 13, 16, 23–25, 31, 150 Aristotle’s view on motion, 13, 17, 22, 30 atom, radioactive, 167 atomism, 162 attraction electric forces of, 263 gravitational, 243, 262 Bell, J., 136 block universe, 122 Bohr, N., 157, 178 Born, M., 160 causality, relativistic, 124 charge electric, 189 inertial, 264 non-inertial, 263, 264 strong, 264 weak, 264 classical mechanics, 181 clock dimensionality of, 128 four-dimensional, 140 three-dimensional, 127, 133, 141 Compton frequency, 162, 166 conductivity, electrical, 173 consciousness, 146, 149, 150 implicitly defined as an entity, 150 consciousnessless, 151 bodies, 152 conventionality of simultaneity, 146 of what exists, 147 Copernican revolution, 13 Copernican system, 30 Copernicus, N., 13, 16, 17 correspondence principle, 178, 180 different aspects of, 178 epistemological aspect of, 179, 180 ontological aspect of, 179, 182 Coulomb field, 256, 258, 260 cross-section, three-dimensional, 137 deduction, hidden, 172 delayed discovery, 52 descriptions of the world four-dimensional, 124 three-dimensional, 124 detector, 167 Dichotomy, 162 Dieks, D., 124, 147 288 Index dimensionality, 124 dipole moment, 161 average value of, 161 discreteness in space and time, 162 Dostoevsky’s method of cruel experimentation, 181 double-slit experiment, 165 Einstein, A., 133, 149 Einstein, A., 48, 159 Eleatic school of philosophy, 14 Eleatics, 14 electrodynamics, 48 electromagnetic disturbance, 189 electron, 160 accelerated, 261 as a particle, 161 classical, 253, 256, 260–262 classical model of, 254 disintegrated worldline of, 162 inertial, 256 measured as localized particle, 160 point-like particle, 160 radiating, 254 some kind of fluid, 160 energy kinetic, 113 relativistic, 114 rest, 113 equivalence principle, 208, 211, 225, 257, 261 event, 56 empty, 57 event solipsism, 124 events obectively divided into past, present, and future, 148 pre-relativistic division of, 123, 124, 133, 137 existence absolute, 125, 130, 141 absoluteness of, 125 continuous in time, 159 discontinuous in time, 159 frame-dependent, 130 frame-independent, 125, 130 observer-dependent, 130 observer-independent, 125, 130 ontological relativization of, 131, 142 ontologically relativized, 126, 128, 139, 142 relative, 130 relativization of, 142 relativized, 125, 130, 132 experiment Gedanken, 39 mechanical, 46 thought, 39 ultimate judge, 46 experimental results cannot contradict one another, 119 Fermi, E., 225, 227, 252 Feynman, R., 160, 228, 253 field distorted, 256, 258, 262, 264 electric, 260 instantaneous, 256 electromagnetic quantized, 263 gravitational, 188, 260, 261, 265 magnetic, 167 undistorted, 263 fifth force, 264 fifth interaction, 264 Fitzgerald, G., 136 flow of time, 146 mind-dependent, 150, 151 objective, 148, 150 force external, 243, 247, 259 fictitious, 244, 245 gravitational, 188, 243, 258, 264, 265 inertial, 189, 243, 246, 256, 264 non-gravitational nature of, 259 radiation reaction, 254 real, 245 resistance, 265 Index restoring, 188, 246 4-atom, 163 4-atomism, 164, 166 four-dimensionalist view, 48, 127, 133 four-dimensionality, 189 four-momentum, 114, 115 four-vector, 115 displacement, 115 velocity, 112 four-velocity, 112 frame accelerated, 261 rest, 112 free will, 146, 152 future, 137 ontologically undetermined, 152 open, 159 Galileo’s principle of relativity, 11, 26, 29–31, 37, 40, 42, 43, 45–48, 50, 51, 53, 55, 121 physical meaning of, 38 Galileo, G., 9, 13, 15, 17, 25, 26, 30, 31, 39, 43, 46, 47 general relativity, 121, 155, 156, 243, 259, 260, 269 genetics, 174 geocentric system, 13 geodesic path, 243, 258, 264, 265 geometry Euclidean, 142 pseudo-Euclidean, 82, 83, 104, 142 gluon, 210, 264, 265 gravitational interaction, mechanism of, 243 gravity, 180, 243 force of, 243 mystery of, 243 Griffiths, D., 228 ground state, 160 Heaviside, O., 252 Hegel, G., 32 heliocentric model, 18 289 heliocentric system, 13, 16 hidden deduction, 172 Higgs force, 264 Hinton, C.H., 56 Hume’s problem, 171 Hume, D., 172 hydrogen atom, 160 hypothetico-deductive method, 13, 172 incommensurability, 180 induction, 172, 176 justification of, 171, 172, 176 inertia, 46, 47, 61, 63, 191, 243, 252, 265 as resistance, 243 electromagnetic, 253 open question of, 189 origin of, 188 unsolved mystery, 243 inertial force, 188, 246 as restoring force, 246 reality of, 244 inference deductive, 171, 173, 181, 182 hidden deductive, 182 inductive, 171, 181, 182 incorrect, 173, 177 probabilistic, 176 information-collecting sphere, 227 interactions electromagnetic, 211, 260, 265 strong, 211, 260, 264, 265 weak, 211, 260, 264, 265 internal logic, 29, 55 of fundamental ideas, 29, 35, 50, 52 Kepler, J., 13 knowledge, absolute, 171 Kwal, B., 252 language four-dimensional, 187 three-dimensional, 50, 121, 125, 128, 132, 134, 164, 187 290 Index two-dimensional, 187 Leibnitz, G., 31 length contraction, 121, 135, 138, 156 true explanation of, 136 length, proper, 134 level macroscopic, 126, 178 microscopic, 158, 178 quantum, 180 Li´enard–Wiechert potentials, 227 light anisotropic propagation of, 189, 227 anisotropic velocity of, 189 average anisotropic velocity of, 256, 262 average coordinate velocity of, 201 average isotropic velocity of, 260 average proper velocity of, 202, 227, 262 light cone future, 66 past, 66 light wave spherical, 39 expanding, 40 Lorentz invariance, 166 Lorentz, H., 252 Lowe, E.J., 124 luminiferous ether, 49 MacGregor, M., 253 Mach, E., 252 Madelung, E., 161 Mandel, H., 252 many spaces, 38, 43, 51 three-dimensional, 45 mass active gravitational, 261, 262 electromagnetic, 252, 262 inertial, 243, 253, 261, 265 as measure of resistance, 243 passive gravitational, 116, 244, 258, 261, 265 as measure of resistance, 244 proper, 115 relativistic, 114, 115 relativistic increase of, 114 rest, 112, 115 matter, 243 Maxwell’s electrodynamics, 256 Maxwell, N., 123 McCall, S., 124 metric, Schwarzschild, 212 Michelson–Morley experiment, 46 Minkowski spacetime, 121–123, 125, 142, 146, 187, 188 ontological status of, 122 Minkowski world, 148 Minkowski, H., 9, 30, 51, 56, 98, 149, 156, 187 motion absolute, 31, 45, 55, 58 non-existence of, 58 absolute uniform, 47 accelerated, 260 by inertia, 258, 259 non-resistant, 259 relative, 125, 130, 132, 137 motionless Earth, 18 moving Earth, 18 muon experiment, 135 Nature does not conspire against us, 47 Newton’s second law, 256, 271 Newton’s third law, 256 Newton, I., 31 Newtonian mechanics, 159, 178, 180 now, 44, 148 global, 148 local, 149 nucleus, 161 object fluid-like, 165 four-dimensional, 127, 128, 130, 132, 137, 138, 140, 148 geometrical, 56 Index indivisible, structured in time, 166 physical, 56, 127 structureless, structured in time, 166 three-dimensional, 127, 132, 137, 140 extended, 137 objective probabilism, 159 objectively probabilistic behaviour, 159 off-mass-shell particles, 266 ontology four-dimensional, 124 three-dimensional, 124 Panofsky, W., 227 particle inertial, 265 non-inertial, 189, 244, 247 past, 137 Petrov, S., 29 Phillips, M., 228 photon normal, 267 inertia of, 271 virtual, 263, 267, 268 physical laws macroscopic, 179 quantum, 179 Planck’s constant, 179 Poincar´e, H., 252 Popper, K., 171, 177 present, 44, 122, 124, 125, 137 the only reality, 126 presentism, 44, 123–125, 128, 139 relativized version of, 130, 133, 142 traditional, manifestly wrong, 141 presentists, 127 presents, 125 different, 44 principle of relativity, 29 probability epistemological, 159 objective, 159 291 ontological, 159 wave of, 160 proper length, 92 proper time, 88, 140, 141, 143, 144, 156 proton, 161 Pryce, M., 252 Ptolemaic system, 16, 18 Ptolemy, 13, 14 Putnam, H., 123 Pythagorean theorem, 83 pseudo-Euclidean version, 83 quanton, 162 spin of, 167 quantum electrodynamics, 267 quantum entanglement, 166 quantum field theory, 262 quantum gravity, 155, 156 quantum mechanics, 155–157, 178, 180 Copenhagen interpretation, 160 probabilistic laws of, 157 probabilistic theory, 157 standard interpretation of, 160 quantum object, 160, 161, 166 always measured as localized particle, 161 exists discontinuously in time, 162 nature of, 164 real collapse of, 165 quantum paradox, 162, 164 Răomer, O., 35, 39, 44 reasoning deductive, 171, 173 hidden deductive, 174 inductive, 171, 176 correct, 172 incorrect, 173 recoils, 265 redshift, gravitational, 156, 213 reference frame, 59 inertial, 62, 140 comoving, 62 292 Index instantaneous, 62 non-inertial, 189, 201, 246 relationism, 31 relativistic effects, 187 relativity principle, 51, 105, 140, 144, 252 meaning of, 32 relativity, 4D formulation of, 48 repulsion, electric forces of, 263 rest, absolute, 31 Riemann tensor, 213 Rietdijk, C.W., 123 Rietdijk–Putnam–Maxwell argument, 123 rod three-dimensional, 188 deformed, 189 Rohrlich, F., 252 Sagredo, 18, 23, 24 Saint Augustine, 69, 150 Salviati, 18, 22 scale atomic, 179 macroscopic, 155, 168 microscopic, 158 Schră odinger cat paradox, 166, 167 Schwartz, M., 228 Schwarzschild metric, 212 scientific knowledge, reliability of, 119, 171 scientific theories confirmation of, 177 validity of, 177 Searle, G., 252 self-force, 251, 255, 257, 263, 265 Shapiro time delay, 211 Simplicio, 18, 22, 24 simultaneity absolute, 42, 125 absoluteness of, 45 not absolute, 38 relativity of, 125, 128, 134, 142, 156 contradiction with, 125 Sommerfeld, A., 51 space absolute, 31 as an entity, 31 contraction of, 135 Euclidean, 11 extra-dimensional, 43 four-dimensional, 42, 44, 56 frozen, 45 mathematical, 187 nature of, 31 three-dimensional, 35, 59 instantaneous, 140 single, 42 three-dimensionality of, 43 spacetime, 57, 58, 243 absolutely existing, 125 curved, 259 debate on the nature of, 156 existence of, 158 flat, 191, 259–261 nature of, 155, 164, 178, 180, 189 pseudo-Euclidean nature of, 89, 98, 146 reality of, 119, 121, 155, 156, 171, 177, 187 spacetime curvature, 180 special relativity, 11, 30, 48, 51, 52, 119, 134, 142, 149, 155–157, 177, 178, 182, 191, 259 consequences of, 158 four-dimensional formulation, 121 kinematic consequences of, 119, 156, 187 validity of, 155 speed of light, constancy of, 50 spin, 167 Standard Model, 262–265 state eternal, 151 God-like, 151 superpositional, 167 Stein, H., 123, 133 strain tensor, 247 Index stress four-dimensional, 189, 246, 263 intolerable, 136 three-dimensional, 189 stress tensor, 135 four-dimensional, 247 string theory, 155, 156 substantivalism, 31 Taylor, E.F., 70 temporal becoming, 146 theory area of applicability of, 178 final, 182 three-dimensionalism, 125, 159 three-dimensionalist, 127 three-dimensionality, 123, 125 time as the fourth dimension, 45 coordinate, 115, 128 proper, 111, 115, 128, 129, 140, 211, 213 time dilation, 88, 121, 136, 139, 141, 156 transformation of particles, 166 translatability, 180 triangle inequality, 99–101, 103 in Euclidean space, 99 in spacetime, 103, 104, 146 twin paradox, 121, 142, 146, 151, 156 two light spheres, 45 velocity of light, average proper, 211 velocity, anisotropic, 202 virtual particles, 161, 166 virtual quanta, 264, 265, 271 anisotropy in the propagation of, 265 volume element, anisotropic, 254, 257 W boson, 264, 265 293 wave function, 163 collapse of, 165 wave–particle duality, 160, 161 Weyl, H., 149–151 Wheeler, J.A., 70 Wilson, W., 252 world curved four-dimensional, 156 dimensionality of, 44, 126, 187 external, 45, 48 four-dimensional, 11, 44, 45, 47, 48, 50, 51, 119, 122, 123, 127, 147, 149 four-dimensionality of, 187 frozen, 45, 58 macroscopic, 179 macroscopic level of, 150 multi-dimensional, 48 probabilistic, 157 real four-dimensional, 156 three-dimensional, 44, 45, 122, 124, 126, 147, 187 timelessly existing, 45, 122 worldline, 11, 56, 58, 139, 191 as a time ruler, 140 curved, 61 geodesic, 116, 258–260, 263, 264 light-like, 70, 82 non-geodesic, 259 space-like, 70 straight, 60 time-like, 70, 115 worldtube, 57, 116, 134, 138, 243, 245, 246 deformed, 188, 245, 263 geodesic, 245 non-deformed, 188 not geodesic, 188 of our body, 150 real four-dimensional object, 188 Z boson, 264–266 Zeno’s paradox, 162 ... to Prof Tzvetan Bonchev (at the time Dean of Sofia University’s Faculty of Physics and Chair of the Department of Atomic Physics) and Prof Ivanka Apostolova (at the time Chair of Sofia University’s... On the Nature of Spacetime – Conceptual and Philosophical Issues is the most provocative of the three parts of the book But it had to be written since the issues raised by the theory of relativity. .. sea, all the things outside seem to the voyagers to be moving in a movement which is the image of their own, and they think on the contrary that they themselves and all the things with them are