A Classical Electromagnetic Mass Theory and the Arguments Against It

According to the classical electromagnetic mass theory, it is the unbalanced repulsion of the volume elements of the charge of an accelerating electron caused by its distorted field that gives rise to the electron’s inertia and inertial mass. Since the electric field of an inertial electron (represented by a straight worldline in flat spacetime) is the Coulomb field, the repulsion of its charge elements cancels out exactly and there is no net force acting on the electron. However, if the electron is accelerated, its field distorts, the balance in the repulsion of its volume elements gets disturbed, and as a result it experiences a net self-force F self which resists its acceleration – it is this resistance that the classical electromagnetic mass theory regards as the electron’s inertia. The self-force opposes the external force that accelerates the electron (i.e., its direction is opposite to the electron’s acceleration a) and turns out a − a to be proportional to a: F self = m a, where the coefficient of propor- tionality ma represents the inertial mass of the electron and is equal to U/c2,whereU is the energy of the electron field; therefore the electron inertial mass is electromagnetic in origin. The electromagnetic mass of the classical electron can be calculated by three independent methods [114]: 2 • Energy-derived electromagnetic mass mU = U/c ,whereU is the field energy of an electron at rest. [When the electron is 2 moving with relativistic velocities v,thenmU = U/γc ,where − / γ = 1 − v2/c2 1 2.] • Momentum-derived electromagnetic mass mp = p/v,wherep is the field momentum when the electron is moving at speed v.(For relativistic velocities mp = p/γv.) • Self-force-derived electromagnetic mass ms = Fself /a,whereFself is the self-force acting on the electron when it has acceleration a.(For 3 relativistic velocities ms = Fself /γ a.) 274 A Classical Electromagnetic Mass Theory

There have been two arguments against regarding the entire mass of charged particles as electromagnetic in classical (non-quantum) physics: • There is a factor of 4/3 which appears in the momentum-derived and self-force-derived electromagnetic mass – mp =4mU /3and ms =4mU /3. (The energy-derived electromagnetic mass mU does not contain that factor.) Obviously, the three types of electromagnetic mass should be equal. • The inertia and mass of the classical electron originate from the unbalanced mutual repulsion of its ‘parts’ caused by the distorted electric field of the electron. However, it is not clear what maintains the electron stable since the classical model of the electron describes its charge as uniformly distributed on a spherical shell, and this means that its volume elements tend to blow up since they repel one another. Feynman considered the 4/3 factor in the electromagnetic mass expression a serious problem since it made the electromagnetic mass theory (yielding an incorrect relation between energy and momentum due to the 4/3 factor) inconsistent with the special theory of relativity [74, p. 28-4]: “It is therefore impossible to get all the mass to be electromagnetic in the way we hoped. It is not a legal theory if we have nothing but electrodynamics.” It seems he was unaware that the 4/3 factor which appears in the momentum-derived electromagnetic mass had already been accounted for in the works of Mandel [96], Wilson [97], Pryce [98], Kwal [99], and Rohrlich [100]. (Each of them removed that factor in- dependently from one another.) The self-force-derived electromagnetic mass has been the most difficult to deal with, persistently yielding the factor of 4/3 [114]. By a covariant application of the Hamilton principle in 1921, Fermi [71] first indirectly showed that there was no 4/3 factor in the self-force acting on a charge supported in a gravitational field. In Chap. 10, we saw how the factor of 4/3 is accounted for in the case of an electron at rest in an accelerating reference frame N a and in a frame N g at rest in a gravitational field of strength g, described in N a and N g, respectively. After the 4/3 factor has been removed, the electromagnetic mass theory of the classical electron becomes fully consistent with relativity and the classical electron mass turns out to be purely electromagnetic in origin. Since its origin a century ago, the electromagnetic mass theory has not been able to explain why the electron is stable (what holds its charge together). This failure has been seen as related to the pres- A Classical Electromagnetic Mass Theory 275 ence of the 4/3 factor and has been used as evidence against regarding its entire mass as electromagnetic. To account for the 4/3 factor, it had been assumed that part of the electron mass (regarded as mechanical) originated from non-electric forces (known as the Poincaré stresses [91, 92]) which hold the electron charge together. It was the inclusion of those forces in the classical electron model and the result- ing mechanical mass that compensated the 4/3 factor by reducing the momentum-derived electromagnetic mass from (4/3)mU to mU .This demonstrates that the attractive non-electric Poincaré forces make a negative contribution to the entire electron mass. It turned out, however, that the 4/3 factor was a result of incorrect calculations of the momentum-derived electromagnetic mass as shown by Mandel [96], Wilson [97], Pryce [98], Kwal [99], and Rohrlich [100]. As there re- mained nothing to be compensated (in terms of mass), if there were some unknown attractive forces responsible for holding the electron charge together, their negative contribution (as attractive forces) to the electron mass would reduce it from mU to (2/3)mU . Obviously, there are two options in such a situation – either to seek what this time compensates the negative contribution of the Poincaré stresses to the mass or to assume that the hypothesis of their existence was not necessary in the first place (especially after it turned out that the 4/3 factor does not appear in the correct calculation of the momentum-derived electromagnetic mass). A strong argument supporting the latter option is the fact that, if there existed a real problem with the stability of the electron, the hypothesis of the Poincaré stresses would be needed to balance the mutual repulsion of the volume elements not only of an electron moving with constant velocity (as in the case of the momentum-derived electromagnetic mass), but also of an electron at rest in its rest frame. This, however, is not the case since when the electron is at rest, there is no 4/3 factor problem in the rest-energy-derived electromagnetic mass of the electron. If the electron charge tended to blow up as a result of the mutual repulsion of its ‘parts’, it should do so not only when it is moving at constant velocity but also when it as at rest. Somehow this obvious argument has been overlooked. Another indication that the stability problem does not appear to be a real problem is that it does not show up (through the 4/3 factor) in the correct calculations of the self-force either. As Fermi [71] showed, and as we have seen in Chap. 10, the Poincaré stresses are not needed for the derivation of the self-force-derived electromagnetic mass since the 4/3 factor which was present in previous derivations of the self- 276 A Classical Electromagnetic Mass Theory force turned out to be a result of not including in the calculations an anisotropic volume element which arises due to the anisotropic velocity of light in the non-inertial reference frames where the self-force is calculated. All this implies that there is no real problem with the stability of the electron. We do not know why. What we do know, however, is that, if there were a stability problem, it would inevitably show up in all calculations of the energy-derived, momentum-derived, and self- force-derived electromagnetic mass, which is not the case. Obviously, there should be an answer to the question of why calculations based on the indisputably wrong classical model of the electron (i) correctly describe its inertial and gravitational behaviour (including the equivalence of its inertial and passive gravitational mass), and (ii) yield the correct expressions for the inertial and gravitational force. My guess of what that answer might be is that there is something in the classical model which leads to the correct results. Most probably, the spherical distribution of the charge. However, it is not necessary to assume that this distribution is a solid spherical shell existing at every single in- stant as a solid shell (like the macroscopic objects we are aware of) – only in that case its ‘parts’ will repel one another and the sphere will tend to explode. As we have seen in Chap. 6, it is not unthinkable to picture an elementary charge as being spherical but not solid (with a continuous distribution of the charge). The fact that the 4/3 factor has been accounted for and the stability problem does not appear to be a real problem (since it does not show up either in the rest-energy-derived electromagnetic mass of the electron, or in the calculations of the self-force) indicates that, in the case of the classical electron, the arguments against regarding its inertia and inertial mass as entirely electromagnetic in origin are answered. B Calculation of the Self-Force

The self-force · · a 1 n a n − 1 a r 2 F self = 2 + 2 n 2 a 1+ 2 ρ dV dV1 4π0 r c r c r 2c can be written (to within terms proportional to c−2)as · a 1 n 3 a n − 1 2 F self = 2 + 2 n 2 a ρ dV dV1 . (B.1) 4π0 r 2 c r c r We have reached this result assuming that the charge element dea acts a a upon the charge element de1. In this case the vector r begins at de a a a a and ends at de1, i.e., n points from de to de1.Ifweassumedthatde1 acted upon dea, the result should be the same. As interchanging the two charge elements reverses the direction of n, the self-force in this case will be · a 1 − n 3 a n − 1 2 F self = 2 + 2 n 2 a ρ dV dV1 . (B.2) 4π0 r 2 c r c r Adding equations (B.2) and (B.1) and dividing the result by 2, we get · a 1 3 a n − 1 2 F self = 2 n 2 a ρ dV dV1 . (B.3) 4π0 2 c r c r In order to do the integral (B.3), let us consider the integral [110] a · n I = n dV dV . (B.4) r 1

We can put n = n + n⊥,wheren is parallel to a and n⊥ is perpen- dicular to a.Then 278 B Calculation of the Self-Force (a · n)n = a· n + n⊥ n + n⊥ = a · n + a · n⊥ n + n⊥ = a · n n + a · n n⊥ +(a · n⊥) n +(a · n⊥) n⊥ = a · n n + a · n n⊥ , since (a · n⊥) = 0. Substituting this result in (B.4) yields a · n a · n I = n dV dV + n⊥ dV dV . (B.5) r 1 r 1 To facilitate the calculations further, let us assume that r is rotated 180◦ about an axis parallel to a running through the centre of the spherical charge distribution of the electron. Then the vector n = n + n⊥ becomes n − n⊥. This means that, in the second integral in (B.5), for every elementary contribution a · n n⊥ dV dV , r 1 there is also an equal and opposite contribution a · n − n⊥ dV dV , r 1 which shows that the second integral in (B.5) is zero and we can write a · n I = n dV dV . (B.6) r 1

The integral I is now a function of n alone. In order to return to the general case of n (and not restrict ourselves to using n), we will express the integral in (B.6) in terms of n and a unit vector u in the direction of a.Sincen is parallel to a,wehavea · n = an.Then we can write 2 2 2 (an)n = a(n) = a 1 n 2 2 2 = a 1n = a un = a(un cos θ)

= a(u · n)2 , where θ is the angle the vector n forms with the acceleration vector a. Now we can write the integral (B.6) in the form B Calculation of the Self-Force 279 (u · n)2 I = a dV dV . (B.7) r 1 Following Abraham [88] and Lorentz [90], we have assumed a spher- ically symmetric distribution of the electron charge. This shows that, as all directions in space are indistinguishable, the integral in (B.7) should be independent of the direction of the unit vector u. Hence the average of this integral over all possible directions of u should be equal to the integral itself: (u · n)2 1 (u · n)2 dV dV = dΩ dV dV (B.8) r 1 4π r 1 1 dV dV = 1 (u · n)2dΩ, 4π r where dΩ is the element of solid angle within which a given unit vector u lies. To do this integral, we choose a polar coordinate system with polar axis along n.Thenu · n =cosθ and dΩ =sinθdθdϕ and 1 1 π 2π (u · n)2dΩ = cos2 θ sin θdθ dϕ 4π 4π 0 0 1 π = cos2 θ sin θdθ 2 0 π 1 1 3 = − cos θ 2 3 0 1 1 = − (−1 − 1) 2 3 1 = . 3 Substituting this result into (B.8) yields (u · n)2 1 dV dV dV dV = 1 . r 1 3 r Thus for the integral (B.7), we have (u · n)2 a dV dV I = a dV dV = 1 . (B.9) r 1 3 r Substituting (B.9) into (B.3), we obtain 280 B Calculation of the Self-Force a 1 3 a − a 2 F self = 2 2 ρ dV dV1 4π0 2 3c r c r 2 − a ρ = 2 dV dV1 , 8π0c r and ﬁnally the expression for the self-force becomes U F a = − a . self c2 References

1. H. Weyl: Philosophy of Mathematics and Natural Science (Princeton University Press, Princeton 1949) p. 116 2. 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) 3. 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) 4. G. Galileo: Dialogue Concerning the Two Chief World Systems – Ptole- maic and Copernican, 2nd edn. (University of California Press, Berke- ley 1967) 5. J. Barnes: Early Greek Philosophy, 2nd edn. (Penguin Books, London 2001) Part II 6. J. Barnes: The Presocratic Philosophers (Routledge, London, New York 1982) Chap. X 7. Aristotle: Physics.In:Great Books of the Western World,Vol.7,ed. by M.J. Adler (Encyclopedia Britannica, Chicago 1993) 8. 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 9. 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. 4 14. A.P. French: Special Relativity (Norton, New York, London 1968) p. 72 15. R. Resnick, D. Halliday: Basic Concepts in Relativity and Early Quan- tum Theory, 2nd edn. (Macmillan, New York 1992) p. 27 16. U.E. Schr¨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, 5 (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 Univer- sity Press, Cambridge 2002) 33. S. Saunders: How Relativity Contradicts Presentism. In: [32] 34. S. Savitt: Phil. Sci. 67 (Proceedings), S563 (2000) 35. T. Sider: Four-Dimensionalism. An Ontology of Persistence and Time (Clarendon Press, Oxford 2001) 36. B. Rossi, D.B. Hall: Phys. Rev. 57, 223 (1941) 37. J.B. Hartle: Gravity: An Introduction to Einstein’s General Relativity (Addison Wesley, San Francisco 2003) 38. J. S. Bell: Speakable and Unspeakable in Quantum Mechanics (Cam- bridge University Press, Cambridge 1987) p. 67 39. K. G¨odel: A Remark about the Relationship Between Relativity and Idealistic Philosophy. In: Albert Einstein: Philosopher–Scientist,ed.by P. Schilpp (Open Court, La Salle 1949) p. 558 40. V. Petkov: The Flow of Time According to Eleatic Philosophy and the Theory of Relativity. In: Structur und Dynamik wissenschaftlicher Theorien, ed. by C. Toegel (P. Lang, Frankfurm am Main Bern New York 1986) pp. 121–149 41. C.W. Misner, K.S. Thorne, J.A. Wheeler: Gravitation (Freeman, San Francisco 1973) 42. R.A. Mould, Basic Relativity (Springer, Berlin, Heidelberg, New York 1994) References 283

43. R. d’Inverno: Introducing Einstein’s Relativity (Clarendon Press, Ox- ford 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: Rel- ativity 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’Université de Sofia, St. Kliment Ohrid- ski, Faculté 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. 2 (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 Gen- eral 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 Mag- netism, 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. 2 (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: Relativitäts Theorie, 3rd edn., Vol. 1 (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, Amster- dam 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. Poincaré: Compt. Rend. 140, 1504 (1905) 92. H. Poincaré: Rendiconti del Circolo Matematico di Palermo 21, 129 (1906) 93. E. Fermi: Phys. Zeits. 23, 340 (1922) References 285

94. E. Fermi: Rend. Acc. Lincei (5) 31, 184; 306 (1922) 95. E. Fermi: Nuovo Cimento 25, 159 (1923) 96. H. Mandel: Z. Physik 39, 40 (1926) 97. W. Wilson: Proc. Phys. Soc. 48, 736 (1936) 98. M.H.L. Pryce: Proc. Roy. Soc. A 168, 389 (1938) 99. B. Kwal: J. Phys. Rad. 10, 103 (1949) 100. F. Rohrlich: Am. J. Phys. 28, 639 (1960) 101. M. Jammer: Concepts of Mass in Classical and Modern Physics (Dover, New York, 1997), Chap. 11. See also [102] 102. M. Jammer: Concepts of Mass in Contemporary Physics and Philoso- phy (Princeton University Press, Princeton, 2000) p. 34 103. J.W. Butler: Am. J. Phys. 37, 1258 (1969) 104. E. Mach: Science of Mechanics, 9th edn. (Open Court, London 1933) 105. P. Pearle: Classical Electron Models. In: Electromagnetism: Paths to Research, ed. by D. Teplitz (Plenum Press, New York 1982) pp. 211– 295 106. D. Bender et al: Phys. Rev. D30, 515 (1984) 107. M.H. MacGregor: The Enigmatic Electron (Kluwer, Dordrecht 1992) 108. D. Hestenes, A. Weingartshofer (Eds.): The Electron: New Theory and Experiment (Kluwer, Dordrecht 1991) 109. M. Springford (Ed.): Electron: A Centenary Volume (Cambridge Uni- versity Press, Cambridge 1997) 110. B. Podolsky, K.S. Kunz: Fundamentals of Electrodynamics (Marcel Dekker, New York 1969) p. 288 111. M.K. Gaillard, P.D. Grannis, and F.J. Sciulli: Rev. Mod. Phys. 71, No. 2 S96 (1999) 112. A. Einstein: Ann. Phys. 20, 627 (1906) 113. D.L. Livesey: Atomic and Nuclear Physics (Blaisdell, Massachusetts 1966) p. 117 114. D. J. Griﬃths and R. E. Owen: Am. J. Phys. 51, 1120 (1983) Index

3D/4D dilemma, 123, 124, 142 non-inertial, 263, 264 4/3 factor, 255, 258, 274, 275 strong, 264 weak, 264 Abraham, M., 252 classical mechanics, 181 absolute motion, 49 clock does not exist, 50 dimensionality of, 128 absolute rest, 49 four-dimensional, 140 absolute space, 49 three-dimensional, 127, 133, 141 absolutism, 31 Compton frequency, 162, 166 acceleration, absoluteness of, 189, conductivity, electrical, 173 191 consciousness, 146, 149, 150 Anastassov, A., 162 implicitly defined as an entity, 150 area of applicability of scientific consciousnessless, 151 theory, 119 bodies, 152 Aristarchus of Samos, 13, 16 conventionality Aristotelian doctrine of motion, 17, of simultaneity, 146 24 of what exists, 147 Aristotle, 13, 16, 23–25, 31, 150 Copernican revolution, 13 Aristotle’s view on motion, 13, 17, Copernican system, 30 22, 30 Copernicus, N., 13, 16, 17 atom, radioactive, 167 correspondence principle, 178, 180 atomism, 162 different aspects of, 178 attraction epistemological aspect of, 179, 180 electric forces of, 263 ontological aspect of, 179, 182 gravitational, 243, 262 Coulomb field, 256, 258, 260 cross-section, three-dimensional, 137 Bell, J., 136 block universe, 122 deduction, hidden, 172 Bohr, N., 157, 178 delayed discovery, 52 Born, M., 160 descriptions of the world four-dimensional, 124 causality, relativistic, 124 three-dimensional, 124 charge detector, 167 electric, 189 Dichotomy, 162 inertial, 264 Dieks, D., 124, 147 288 Index dimensionality, 124 frame-independent, 125, 130 dipole moment, 161 observer-dependent, 130 average value of, 161 observer-independent, 125, 130 discreteness in space and time, 162 ontological relativization of, 131, Dostoevsky’s method of cruel 142 experimentation, 181 ontologically relativized, 126, 128, double-slit experiment, 165 139, 142 relative, 130 Einstein, A., 133, 149 relativization of, 142 Einstein, A., 48, 159 relativized, 125, 130, 132 Eleatic school of philosophy, 14 experiment Eleatics, 14 Gedanken, 39 electrodynamics, 48 mechanical, 46 electromagnetic disturbance, 189 thought, 39 electron, 160 ultimate judge, 46 accelerated, 261 experimental results cannot as a particle, 161 contradict one another, 119 classical, 253, 256, 260–262 classical model of, 254 Fermi, E., 225, 227, 252 disintegrated worldline of, 162 Feynman, R., 160, 228, 253 inertial, 256 field measured as localized particle, 160 distorted, 256, 258, 262, 264 point-like particle, 160 electric, 260 radiating, 254 instantaneous, 256 some kind of fluid, 160 electromagnetic energy quantized, 263 kinetic, 113 gravitational, 188, 260, 261, 265 relativistic, 114 magnetic, 167 rest, 113 undistorted, 263 equivalence principle, 208, 211, 225, fifth force, 264 257, 261 fifth interaction, 264 event, 56 Fitzgerald, G., 136 empty, 57 flow of time, 146 event solipsism, 124 mind-dependent, 150, 151 events objective, 148, 150 obectively divided into past, force present, and future, 148 external, 243, 247, 259 pre-relativistic division of, 123, fictitious, 244, 245 124, 133, 137 gravitational, 188, 243, 258, 264, existence 265 absolute, 125, 130, 141 inertial, 189, 243, 246, 256, 264 absoluteness of, 125 non-gravitational nature of, 259 continuous in time, 159 radiation reaction, 254 discontinuous in time, 159 real, 245 frame-dependent, 130 resistance, 265 Index 289

restoring, 188, 246 heliocentric system, 13, 16 4-atom, 163 hidden deduction, 172 4-atomism, 164, 166 Higgs force, 264 four-dimensionalist view, 48, 127, Hinton, C.H., 56 133 Hume’s problem, 171 four-dimensionality, 189 Hume, D., 172 four-momentum, 114, 115 hydrogen atom, 160 four-vector, 115 hypothetico-deductive method, 13, displacement, 115 172 velocity, 112 four-velocity, 112 incommensurability, 180 frame induction, 172, 176 accelerated, 261 justiﬁcation of, 171, 172, 176 rest, 112 inertia, 46, 47, 61, 63, 191, 243, 252, free will, 146, 152 265 future, 137 as resistance, 243 ontologically undetermined, 152 electromagnetic, 253 open, 159 open question of, 189 origin of, 188 Galileo’s principle of relativity, 11, unsolved mystery, 243 26, 29–31, 37, 40, 42, 43, 45–48, inertial force, 188, 246 50, 51, 53, 55, 121 as restoring force, 246 physical meaning of, 38 reality of, 244 Galileo, G., 9, 13, 15, 17, 25, 26, 30, inference 31, 39, 43, 46, 47 deductive, 171, 173, 181, 182 general relativity, 121, 155, 156, 243, hidden deductive, 182 259, 260, 269 inductive, 171, 181, 182 genetics, 174 incorrect, 173, 177 geocentric system, 13 probabilistic, 176 geodesic path, 243, 258, 264, 265 information-collecting sphere, 227 geometry interactions Euclidean, 142 electromagnetic, 211, 260, 265 pseudo-Euclidean, 82, 83, 104, 142 strong, 211, 260, 264, 265 gluon, 210, 264, 265 weak, 211, 260, 264, 265 gravitational interaction, mechanism internal logic, 29, 55 of, 243 of fundamental ideas, 29, 35, 50, gravity, 180, 243 52 force of, 243 Kepler, J., 13 mystery of, 243 knowledge, absolute, 171 Griﬃths, D., 228 Kwal, B., 252 ground state, 160 language Heaviside, O., 252 four-dimensional, 187 Hegel, G., 32 three-dimensional, 50, 121, 125, heliocentric model, 18 128, 132, 134, 164, 187 290 Index

two-dimensional, 187 as measure of resistance, 244 Leibnitz, G., 31 proper, 115 length contraction, 121, 135, 138, relativistic, 114, 115 156 relativistic increase of, 114 true explanation of, 136 rest, 112, 115 length, proper, 134 matter, 243 level Maxwell’s electrodynamics, 256 macroscopic, 126, 178 Maxwell, N., 123 microscopic, 158, 178 McCall, S., 124 quantum, 180 metric, Schwarzschild, 212 Li´enard–Wiechert potentials, 227 Michelson–Morley experiment, 46 light Minkowski spacetime, 121–123, 125, anisotropic propagation of, 189, 142, 146, 187, 188 227 ontological status of, 122 anisotropic velocity of, 189 Minkowski world, 148 average anisotropic velocity of, Minkowski, H., 9, 30, 51, 56, 98, 149, 256, 262 156, 187 average coordinate velocity of, 201 motion average isotropic velocity of, 260 absolute, 31, 45, 55, 58 average proper velocity of, 202, non-existence of, 58 227, 262 absolute uniform, 47 light cone accelerated, 260 future, 66 by inertia, 258, 259 past, 66 non-resistant, 259 light wave relative, 125, 130, 132, 137 spherical, 39 motionless Earth, 18 expanding, 40 moving Earth, 18 Lorentz invariance, 166 muon experiment, 135 Lorentz, H., 252 Lowe, E.J., 124 Nature does not conspire against us, luminiferous ether, 49 47 Newton’s second law, 256, 271 MacGregor, M., 253 Newton’s third law, 256 Mach, E., 252 Newton, I., 31 Madelung, E., 161 Newtonian mechanics, 159, 178, 180 Mandel, H., 252 now, 44, 148 many spaces, 38, 43, 51 global, 148 three-dimensional, 45 local, 149 mass nucleus, 161 active gravitational, 261, 262 electromagnetic, 252, 262 object inertial, 243, 253, 261, 265 ﬂuid-like, 165 as measure of resistance, 243 four-dimensional, 127, 128, 130, passive gravitational, 116, 244, 132, 137, 138, 140, 148 258, 261, 265 geometrical, 56 Index 291

indivisible, structured in time, 166 ontological, 159 physical, 56, 127 wave of, 160 structureless, structured in time, proper length, 92 166 proper time, 88, 140, 141, 143, 144, three-dimensional, 127, 132, 137, 156 140 proton, 161 extended, 137 Pryce, M., 252 objective probabilism, 159 Ptolemaic system, 16, 18 objectively probabilistic behaviour, Ptolemy, 13, 14 159 Putnam, H., 123 off-mass-shell particles, 266 Pythagorean theorem, 83 ontology pseudo-Euclidean version, 83 four-dimensional, 124 three-dimensional, 124 quanton, 162 spin of, 167 Panofsky, W., 227 quantum electrodynamics, 267 particle quantum entanglement, 166 inertial, 265 quantum field theory, 262 non-inertial, 189, 244, 247 quantum gravity, 155, 156 past, 137 quantum mechanics, 155–157, 178, Petrov, S., 29 180 Phillips, M., 228 Copenhagen interpretation, 160 photon probabilistic laws of, 157 normal, 267 probabilistic theory, 157 inertia of, 271 standard interpretation of, 160 virtual, 263, 267, 268 quantum object, 160, 161, 166 physical laws always measured as localized macroscopic, 179 particle, 161 quantum, 179 exists discontinuously in time, 162 Planck’s constant, 179 nature of, 164 Poincaré, H., 252 real collapse of, 165 Popper, K., 171, 177 quantum paradox, 162, 164 present, 44, 122, 124, 125, 137 the only reality, 126 Römer, O., 35, 39, 44 presentism, 44, 123–125, 128, 139 reasoning relativized version of, 130, 133, deductive, 171, 173 142 hidden deductive, 174 traditional, manifestly wrong, 141 inductive, 171, 176 presentists, 127 correct, 172 presents, 125 incorrect, 173 different, 44 recoils, 265 principle of relativity, 29 redshift, gravitational, 156, 213 probability reference frame, 59 epistemological, 159 inertial, 62, 140 objective, 159 comoving, 62 292 Index

instantaneous, 62 Sommerfeld, A., 51 non-inertial, 189, 201, 246 space relationism, 31 absolute, 31 relativistic effects, 187 as an entity, 31 relativity principle, 51, 105, 140, contraction of, 135 144, 252 Euclidean, 11 meaning of, 32 extra-dimensional, 43 relativity, 4D formulation of, 48 four-dimensional, 42, 44, 56 repulsion, electric forces of, 263 frozen, 45 rest, absolute, 31 mathematical, 187 Riemann tensor, 213 nature of, 31 Rietdijk, C.W., 123 three-dimensional, 35, 59 Rietdijk–Putnam–Maxwell argu- instantaneous, 140 ment, 123 single, 42 rod three-dimensionality of, 43 three-dimensional, 188 spacetime, 57, 58, 243 deformed, 189 absolutely existing, 125 Rohrlich, F., 252 curved, 259 debate on the nature of, 156 Sagredo, 18, 23, 24 existence of, 158 Saint Augustine, 69, 150 Salviati, 18, 22 flat, 191, 259–261 scale nature of, 155, 164, 178, 180, 189 atomic, 179 pseudo-Euclidean nature of, 89, macroscopic, 155, 168 98, 146 microscopic, 158 reality of, 119, 121, 155, 156, 171, Schrödinger cat paradox, 166, 167 177, 187 Schwartz, M., 228 spacetime curvature, 180 Schwarzschild metric, 212 special relativity, 11, 30, 48, 51, 52, scientific knowledge, reliability of, 119, 134, 142, 149, 155–157, 119, 171 177, 178, 182, 191, 259 scientific theories consequences of, 158 confirmation of, 177 four-dimensional formulation, 121 validity of, 177 kinematic consequences of, 119, Searle, G., 252 156, 187 self-force, 251, 255, 257, 263, 265 validity of, 155 Shapiro time delay, 211 speed of light, constancy of, 50 Simplicio, 18, 22, 24 spin, 167 simultaneity Standard Model, 262–265 absolute, 42, 125 state absoluteness of, 45 eternal, 151 not absolute, 38 God-like, 151 relativity of, 125, 128, 134, 142, superpositional, 167 156 Stein, H., 123, 133 contradiction with, 125 strain tensor, 247 Index 293 stress wave function, 163 four-dimensional, 189, 246, 263 collapse of, 165 intolerable, 136 wave–particle duality, 160, 161 three-dimensional, 189 Weyl, H., 149–151 stress tensor, 135 Wheeler, J.A., 70 four-dimensional, 247 Wilson, W., 252 string theory, 155, 156 world substantivalism, 31 curved four-dimensional, 156 dimensionality of, 44, 126, 187 Taylor, E.F., 70 external, 45, 48 temporal becoming, 146 four-dimensional, 11, 44, 45, 47, theory 48, 50, 51, 119, 122, 123, 127, area of applicability of, 178 147, 149 final, 182 four-dimensionality of, 187 three-dimensionalism, 125, 159 frozen, 45, 58 three-dimensionalist, 127 macroscopic, 179 three-dimensionality, 123, 125 macroscopic level of, 150 time multi-dimensional, 48 as the fourth dimension, 45 probabilistic, 157 coordinate, 115, 128 real four-dimensional, 156 proper, 111, 115, 128, 129, 140, three-dimensional, 44, 45, 122, 211, 213 124, 126, 147, 187 time dilation, 88, 121, 136, 139, 141, timelessly existing, 45, 122 156 worldline, 11, 56, 58, 139, 191 transformation of particles, 166 as a time ruler, 140 translatability, 180 curved, 61 triangle inequality, 99–101, 103 geodesic, 116, 258–260, 263, 264 in Euclidean space, 99 light-like, 70, 82 in spacetime, 103, 104, 146 non-geodesic, 259 twin paradox, 121, 142, 146, 151, space-like, 70 156 straight, 60 two light spheres, 45 time-like, 70, 115 worldtube, 57, 116, 134, 138, 243, velocity of light, average proper, 211 245, 246 velocity, anisotropic, 202 deformed, 188, 245, 263 virtual particles, 161, 166 geodesic, 245 virtual quanta, 264, 265, 271 non-deformed, 188 anisotropy in the propagation of, not geodesic, 188 265 of our body, 150 volume element, anisotropic, 254, real four-dimensional object, 188 257 Z boson, 264–266 W boson, 264, 265 Zeno’s paradox, 162