Concepts of Simultaneity: From Antiquity to Einstein and Beyond

13 Clock Transport Synchrony

The use of the transport of clocks for the establishment of distant simultaneity is as old as the invention of portable timepieces. As we saw in chapter 4, transported clocks had been used in the sixteenth century, for example , by the Flemish cartographer Reinerus Gemma, for the determination of the geographic longitude of a certain location. Still in 1904 Henri Poincaré, in his The Value of Science,1 discussed in detail the use of transported clocks for the determination of distant simultaneity and geographical longitude without making any reference to the relativistic retardation involved. In prerelativistic physics, in fact, the transport of clocks poses no problem because according to classical physics motion does not affect the time indication of a clock. In relativistic physics, however, the so-called time dilation, an experimentally well-confirmed effect, impairs the use of moving clocks for the establishment of distant simultaneity unless this time dilation is taken into account. The simplest way to confirm this effect is to synchronize locally two clocks, say U1 and U2, and to move one of them, say U2, away from C H A P T E R T H I R T E E N Clock Transport Synchrony 1H. Poincaré, La Valeur de Science (Paris: Flammarion, 1904); The Value of Science (New York: Dover Publications, 1958), p. 35. its partner and after some time to bring them together again. It will be seen that the two clocks are no longer in synchrony. The well-known “twin paradox ” is a famous example of this effect. Only obstinate opponents of the special theory of relativity, like Hugo Dingler,2 who claimed that motion cannot affect the rate of a clock, saw no problem in the transport of clocks for the establishment of distant synchronization. The retardation of a moving clock U relative to a clock UB, which is at rest in an inertial system S, that is, the amount of time by which U, after leaving UB, lags behind UB when it meets again with UB, had already been calculated by Einstein in § 4 of his 1905 relativity paper. Because of the importance of this effect for our present discussion a presentation of its mathematical derivation is not out of place. Let UA be a clock located at the origin A of an inertial system S and UB a clock located at the point B on the x axis of S and synchronized with UA, and let d denote the distance between these two clocks. Then in S a third clock U, which leaves A at the time t  0 and moves with constant velocity  from A to B, reaches UB at the time t  d/. (13.1) In the inertial system S, in which U is at rest and which is in standard configuration with S, the time required by U to reach B is according to the Lorentz transformation given by t  (1 2/c2) 1/2 (t d/c2). (13.2) By substitution of (13.1) in (13.2) we obtain t  t(1 2/c2)1/2 (13.3) Hence, t # t, (13.4) which means that the arrival time as indicated by U is less than the arrival time as indicated by UB and the difference or retardation of U is given by the expression t t  t t(1 2/c2)1/2  t[1 (1 2/c2)1/2]. (13.5) Clock Transport Synchrony 241 2H. Dingler, “Kritische Bemerkungen zu den Grundlagen der Relativitätstheorie,” Physikalische Zeitschrift 21, 668–675 (1920). 242 Concepts of Simultaneity Neglecting magnitudes of fourth and higher order we conclude that the moving clock lags behind the stationary clock by 1/2(2/c2) seconds per second.3 As this mathematical analysis reveals, the reparation of a moving clock can be reduced arbitrarily by diminishing sufficiently the velocity  of the moving clock. It was for this reason that Joseph Winternitz, a former student of Philipp Frank at the University of Prague, in his critique of Dingler’s rejection of the theory of relativity pointed out that clock transport can serve as a method of synchronization because “one can arbitrarily decrease the retardation effect by sufficiently diminishing the velocity of the clock.”4 At the same time also Eddington declared in his previously mentioned treatise5 that there are two equivalent methods of establishing distant simultaneity : “(1) A clock moved with infinitesimal velocity...