Summary
After drawing some preliminary divisions between different conceptions of time, distinguished as “proper time”, “Kantian time”, “psychological time”, “physical time”, and “mathematical time”, there follows a discussion of time dilatation and the clock paradoxes of Einstein’s restricted theory of relativity. Beginning with Lorentz’s time transformations, and some remarks on the different attitudes of Lorentz, Poincaré, and Einstein to the transformation equations generally, we turn to the clock paradox, often held to involve Einstein in inconsistency. A new notation is introduced, which not only simplifies the manipulation of the Lorentz transformation equations, but which makes it natural to preserve certain terms which are commonly suppressed unknowingly. The notation makes clear what has long been appreciated by some, but denied by others, namely, that the time dilatation equations, correctly expressed, are consistent under a suitable interpretation (and not merely mathematically consistent). Different attempts to engender contradiction within Einstein’s theory are considered, in particular by taking more than two observers, moving uniformly. Necessarily we introduce the notion of an aggregate proper time of two observers, which may not satisfy those looking for a solution to the problem of actual space travellers. There are also difficult problems of specifying what is to count as a journey, under this interpretation. Nevertheless, there is here no sign of the inconsistency in the interpretation of Einstein’s theory for which some writers have argued. Where there is an asymmetry in the aggregate times experienced, there is also an asymmetry in the observers.
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References
Eleven important papers on relativity, including Zur Elektrodynamik bewegter Körper (Einstein, 1905), Raum und Zeit (Minkowski, 1908), and two earlier papers by Lorentz (1895 and 1904) are readily available in English translation in “The Principle of Relativity”, London, Methuen, 1923, reprinted by Dover subsequently. The translation was made from the German edition “Das Relativitätsprinzip”, 4th edition, Teubner, 1922. References below are to the English edition, abbreviated as P. R.
Hermann Weyl’s Raum, Zeit, Materie is available in English translation, unfortunately from the fourth German edition of 1921, rather than the fifth, with the title Space — Time — Matter, London, Methuen, 1921, subsequently reprinted by Dover. For A. S. Eddington, see especially The Mathematical Theory of Relativity, 2nd edition, Cambridge University Press, 1924, p. 26: “the contraction [of a moving rod] and retardation [of a moving clock] do not imply any absolute change in the rod and clock. The ‘configuration of events’ constituting the four-dimensional structure which we call a rod is unaltered; all that happens is that the observer’s space and time partitions cross it in a different direction”.
Kritik der reinen Vernunft, 2nd edition, 1787, I. ii. 7 (b).
Ibid., I. ii. 7 (a).
Ibid., I. ii. 5 (1–2). This did not mean that it had absolute reality in the sense of inhering in things as a condition or property. Ibid., I. ii. 7 (c).
Ibid., I. ii. 7 (b).
Ibid., I. ii. 5 (3).
Whitrow, G. J.: The Natural Philosophy of Time. London: Nelson, 1961, chapter 4.
See note 7.
Theory of conjugate functions, or algebraic couples; with a preliminary and elementary essay on algebra as the science of pure time (read in 1833 and 1835), Transactions of the Royal Irish Academy, Vol. 17 (1837) pp. 293–422. Cf. p. 297: “The notion of intuition of ORDER IN TIME is not less but more deep-seated in the human mind, than the notion or intuition of ORDER IN SPACE ; and a mathematical Science may be founded on the former, as pure and demonstrative as the Science founded on the latter”. Since Hamilton hoped to derive algebra from a prior intuition of time, this was not strictly mathematical in the sense used here.
For an ample bibliography of relativity, see the references given in E. T. Whittaker’s History of the Theories of Aether and Electricity, Vol. 2. London: Nelson 1953, chapter 2. Lorentz’s paper of 1904 (not 1903, as given in error by Whittaker) is printed in P. R.
Aether and Matter, Cambridge University Press, 1900, chapter 11, especially pp. 167–77.
See in particular Astrophysical Journal, Vol. 68 (1928), pp. 385–8, and Lectures on Theoretical Physics, Vol. 3, London: McMillan 1931, pp. 181 to the end. These lectures on relativity date from 1910–12.
Op. cit. (note 11), p. 177. .
Op. cit. (1931, note 12), p. 303.
P. R., pp. 48–50.
That the criticism did not pass unnoticed is suggested by Bishop Barne’s Scientific Theory and Religion, Cambridge University Press, 1933, p. 114, for example.
Naturwiss., Vol. 6 (1918), p. 697.
British Journal for the Philosophy of Science, Vol. 15 (1964), p. 46.
Loc. cit. (note 18).
Nature, Vol. 168 (1951), p. 246.
Scientific Theory and Religion, (The Gifford Lectures, 1927–1929), Cambridge Univerversity Press, 1933, p. 113.
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North, J.D. (1972). The Time Coordinate in Einstein’s Restricted Theory of Relativity. In: Fraser, J.T., Haber, F.C., Müller, G.H. (eds) The Study of Time. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-65387-2_2
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