Abstract
I discuss the historical origins of the idea of the relativity of motion and Einstein’s and Minkowski’s seminal innovations. I then show how attempts to refute the objectivity of time lapse by appeal to relativity theory fail to recognize that time lapse is tracked by Minkowski’s proper time, which is invariant.
The objective status of becoming was strengthened rather than weakened by the special theory of relativity.
—Milič Čapek (1971, 233).
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- 1.
If each is walking at 4 km per hour in opposite directions, the time difference will be about 3 days. Similarly, if I get up from a sitting position and walk at about 3 km h−1 the difference will be about a day (Davies 1995, 70).
- 2.
Here I am ignoring the changes in Huygens’s unpublished views on relativity. He always held to the relativity of rectilinear motions , even in collisions. But by the time of Leibniz’s sojourn in Paris (1672–6), he had come to believe that the centrifugal effect gave a criterion for motion in absolute space , as Newton would later argue. Huygens’s subsequent re-espousal of the relativity of all motion was not the same as Leibniz’s commitment to the Equivalence of Hypotheses concerning appearances, nor did he accept Leibniz’s mature views that true motions could be identified by appeal to causes. See Mormino (1993) and (2011) for full discussion.
- 3.
It should be noted that Galileo assumes all this to be taking place on the spherical surface of the Earth; for him, what requires no cause is the circular motion of a heavy body along the surface of the spherical Earth (assumed frictionless). It was in fact Descartes who proposed that the motion that requires no cause is motion at a constant speed in a straight line. Newton’s animus towards Descartes by the time he wrote the Principia was such that he sought to expunge all record of Descartes’s (significant) influence on his own work.
- 4.
- 5.
As Robert DiSalle observes, “By the later 19th century, observations became sufficiently precise to reveal that there is in fact a leftover acceleration, namely the famous extra precession of Mercury. But that could not affect Newton’s analysis in 1687”. (DiSalle 2009, §2.7)—we will come to the precession of Mercury in Chap. 7 below.
- 6.
Intriguingly, this analysis is completely in accord with Leibniz’s account, save for one important detail. Leibniz had proposed that we adopt as a fiction the hypothesis that a system of bodies (such as the fixed stars) maintain the same mutual situations over time. We then relate positions and changes of position to these bodies using the accepted laws of mechanics, and designate that hypothesis as true which gives the most intelligible description of the phenomena. Adventitious phenomena whose causes remained unexplained on the Ptolemaic hypothesis but explained on the Copernican, for instance, such as the retrograde motions of the planets, the motions of tides and apparent changes of position of distant stars, would serve to confirm that the Copernican hypothesis is the true hypothesis. But Leibniz thought, wrongly, that rotational motion was in itself (i.e. without reference to such phenomena) relative. See Chap. 7 and also Arthur (2013) for further discussion.
- 7.
Einstein himself states the content of the Special Theory of Relativity in the way I am suggesting in his later essay “Physics and Reality” (Einstein 1954, 308): “it is necessary to postulate invariance of all systems of physical equations which express general laws with respect to Lorentz transformations . The elaboration of this requirement forms the content of the special theory of relativity ”. In Appendix A I sketch a group theoretic derivation of special relativity using such a Langean conception of reference frames in preference to the rods and clocks in terms of which Einstein conceived them. The significance of Einstein’s mistaken conception of inertial frames will become clear in Chap. 7, when we discuss Einstein’s path to his General Theory of Relativity .
- 8.
Here we may note that there is an ambiguity in the notion of the relativity of motion . It can mean that the motion of a body is always relative to whichever body is taken to be at rest; or it can mean only that all velocities are relative velocities—i.e. that there are no absolute, instantaneous velocities. But the latter type of relativity does not entail that all motions, including accelerations, are relative to a given reference system. As we shall see in Chap. 7, Einstein regarded the principle that all velocities are relative as yielding only a “restricted” theory of relativity, and sought to provide a properly general theory, in which all motions are relative to reference bodies, however moved.
- 9.
According to Maxwell’s theory, these electromagnetic waves would have a velocity of \(1/ \surd (\upvarepsilon_{0} \upmu_{0} )\), where ε0 = 8.854 × 10−12 F/m and μ0 = 4π × 10−7 H/m are the electric and magnetic constants appearing in his equations, giving a velocity of 2.998 × 108 m/s, the known velocity of light.
- 10.
Ole Rømer (1644–1710), a Danish astronomer working at the Royal Observatory in Paris, determined the velocity of light by timing the eclipses of Io, one of Jupiter’s moons. He announced to the Observatory in a paper of 22 August 1676 that it took “about ten to eleven minutes” for light to traverse “a distance equal to the half-diameter of the terrestrial orbit”, yielding a velocity of about 2.2 × 108 m/s, or about ¾ of the now accepted velocity of light. (His result was accepted by Huygens and Newton, but not fully ratified by astronomers until some decades later.)
- 11.
As emphasized by Roberto Torretti, however, H. A. Lorentz did not follow the British in conceiving electromagnetic waves as involving interaction between ponderable matter and the aether: for him the aether was motionless, in the sense that “no part of it is displaced with respect to its other parts, and that all perceptible motions of the heavenly bodies are motions relative to the aether” (Torretti 1999, 180; quoting Lorentz 1895, from his Collected Papers, vol. 5, p. 4). N.B. I follow Faraday, Maxwell and Torretti in using the British spelling ‘aether’, in order to distinguish this hypothesized medium from the chemical ‘ether’, C2H5OC2H5, a distinction lost with the American spelling.
- 12.
Lorentz defined his local time variable t′ by t′ = t − (v · r)/c2 (Lorentz [1895] 1923).
- 13.
As Poincaré wrote in an extended version of his 1905 note that was published in his (1906), “The reason why we can, without modifying any apparent phenomenon, confer to the whole system a common translation, is that the equations of an electromagnetic medium are not changed under certain transformations which I shall call the Lorentz transformations; two systems, one at rest, the other in translation, thus become exact images of one another”.
- 14.
- 15.
In what follows I shall follow the terminology introduced by Minkowski in his 1908, although he had already articulated the mathematics and physics of his spacetime view in 1907. See (Minkowski 2012) for an English translation of these papers.
- 16.
This is because, as Minkowski relates, “at every worldpoint the expression c2dt2 − dx2 − dy2 − dz2 is always positive, which is equivalent to saying that any velocity v is always less than c” (Minkowski 2012, 115).
- 17.
In his 1907 Minkowski called such a curve the “spacetime line”; he introduced the term “world-line” in his 1908. See Minkowski (2012, 98 and 112, resp.).
- 18.
(Minkowski 2012, 119) Actually, Minkowski had already introduced proper time in this way his lesser-known (1907). There he says the value of the integral of the element dτ “taken along the spacetime line from P0 to a point P is called proper time (Eigenzeit), corresponding to the location of matter at the spacetime point P (This is a generalization of the concept of local time used by Lorentz (Ortzeit) in the case of uniform motion.)” (Minkowski 2012, 99).
- 19.
Isaacson notes Einstein’s preference for the term “Theory of Invariants” (2007, 132), but without giving a reference.
- 20.
- 21.
It should not be thought, however, that it is the acceleration itself that produces the resulting dilation, as opposed to the difference in the length of the spacetime paths resulting from the fact that one twin has taken a path that is at least at some point non-inertial. In fact, Einstein himself was moved (in his 1918) to defend the consistency of special relativistic time dilation by appeal to its consistency with general relativistic time dilation, to be discussed in Chap. 7.
- 22.
An anonymous referee to my (2008) objected that Gödel’s argument depends only on the lapses of time being different for any two arbitrary curves connecting two timelike related events, and that Gödel does not assume that time lapse is measured by a time-coordinate function. But Gödel explicitly construes time lapse in terms of co-ordinate time in his argument from Special Relativity, where his argument against the “relativization of existence” crucially depends on this. This is supported by the interpretation of Yourgrau (1991), who construes Gödel’s argument as depending on a conception of time lapse as relative to reference frame.
- 23.
I use the term ‘degenerate’ here by analogy with quantum theory, where two or more different states may have the same energy level, but may nonetheless be distinguished by the application of an electric or magnetic field.
- 24.
I am indebted to Storrs McCall (private communication) for suggesting to me the relevance here of the analogy with proper length. I am also indebted to Kent Peacock for helping me eradicate some infelicities in my discussion of this in an earlier draft.
- 25.
Quoted from an article on proper length in Wikipedia (http://en.wikipedia.org/wiki/Proper_length: May 5, 2007). The author suggested a generalization of proper length so that it is given by the line integral \({\text{L}} = {\text{c}} \int\nolimits_{\text{P}} {\surd \left[ { -{\text{g}}_{\upmu \upnu } {\text{dx}}^{\upmu } {\text{dx}}^{\text{v} } } \right]}\), where gμν is the metric tensor for the spacetime with +--- signature, normalized to return a time. Since I published this criticism in (Arthur 2008), this Wikipedia article has (as of March 15, 2016) been emended, so that the latter formula is now correctly stated to represent the invariant proper distance along a path, and is correctly distinguished from proper length, “the length of an object in the object’s rest frame”.
- 26.
Minkowski’s judgement is echoed by Einstein in his essay “The Problem of Space, Ether and the Field in Physics”: “Hitherto it had been silently assumed that the four-dimensional continuum of events could be split up into time and space in an objective manner… With the discovery of the relativity of simultaneity , space and time were merged in a single continuum … ” (1954, 281–82).
- 27.
Sklar (1974, 268) correctly points out that, whereas “‘co-ordinate time’ between two events is relative to a given inertial frame”, “[p]roper time is defined only for events at timelike separations and only relative to a particular spacetime curve between the events. On the other hand it is an invariant notion”.
- 28.
(Stein 1968, 11, fn. 6). This quotation from Stein was my starting point for the line of argument developed here. Cf. also p. 16: “… ‘a time co-ordinate’ is not ‘time.’ Neither a nor b is, in any physically significant sense, ‘present’ (or past) for any observer at c—regardless of his velocity—for neither has already become for c (nor has c for them); but a has already become for b, and can influence it” [Here a and b are connectible by a timelike vector ab, the other pairs by spacelike vectors ac and bc.].
- 29.
A particularly striking example is provided by Vesselin Petkov, who, despite clearly recognizing the distinction between Einstein’s relative time and Minkowski’s proper time, regards the view that time flows as “unscientific”. His argument is that if time really flowed then one would have to be able to pick out a unique (global) present moment, “which is the central element of the concept of time flow” (Petkov 2012, 31). We agree that this decisively rules out the classical conception of becoming in terms of a world-wide now; but he fails to acknowledge that this is not injurious to local becoming, where time lapse is tracked by proper time .
- 30.
- 31.
The quotation is from Savitt’s unpublished paper of 2004 (see previous footnote).
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Arthur, R.T.W. (2019). Special Relativity and the Lapse of Time. In: The Reality of Time Flow. The Frontiers Collection. Springer, Cham. https://doi.org/10.1007/978-3-030-15948-1_5
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