A sign error in the Minkowski space-time plot and its consequences

A sign error in the Minkowski space-time plot and its consequences
by J.H.Field, Département de Physique Nucléaire et Corpusculaire Université de Genève.
24, quai Ernest-Ansermet CH-1211 Genève 4. (23. May 2008)


A sign error in an angle while drawing the original Minkowski plot has persisted for a century in text books and the pedagogical literature. When it is corrected, the ‚length contraction‘ effect derived from the geometry of the plot disappears. It is also shown how the ‚relativity of simultaneity‘ effect that has been derived from the plot results from a lack of correspondence between certain geometrical projections on the plot and the properties of the physical system -two spatially separated and synchronised clocks in a common inertial frame  –  that they are purported to describe. 

1 Introduction 
Errors, originating in Einstein’s 1905 special relativity paper [1], in the standard text book interpretation of the physics of the space-time Lorentz transformation (LT), have been pointed out in a series of recent papers by the present author [2, 3, 4]. In these papers it is shown that the ‚length contraction‘ and ‚relativity of simultaneity‘ effects, derived directly from the LT, are spurious, resulting from a failure to include important additive constants in the equations. Einstein pointed out the necessity to include such constants, to correctly describe synchronised clocks, in Ref. [1], but never actually did so himself. Since the demonstration of the spurious nature of the effects is simple, straightforward and brief, it is recalled, for the reader’s convenience, in the following section of the present paper. 

The remainder of this paper is devoted to a discussion of the physics of the Minkowski space-time plot. In his original ’space-time‘ paper [5] Minkowski derived a ‚length contraction‘ effect from the geometry of the plot without considering directly the LT. Similar derivations are to be found in many text books on special relativity or classical electromagnetism.

In Section 3, the projective-geometrical properties of the space-time plot are derived from the LT. In Section 4, Minkowski’s original derivation of ‚length contraction‘ is reviewed, and shown to result from an erroneous assumption concerning the direction of the world line of the considered, uniformly moving, object. In fact, the world line corresponding to Minkowski’s choice of x? and t? axes is x = -vt, whereas it is assumed to be x = vt and therefore to lie along the t? axis. The same mistake is made in all (with, to the present writer’s best knowledge, a single exception) text-book treatments of the problem. Some examples are discussed in Section 5. Also discussed in Section 5 is the fortuitously correct derivation of time dilatation from the standard, incorrect, Minkowski plot, as well as the illusory nature of the ‚relativity of simultaneity‘ effect suggested by superficial inspection of the plot. This error is not, as in the case of ‚length contraction‘, the result of a trivial geometrical mistake, but arises from a naive interpretation of purely mathematical projection operations on the plot that are unrelated to the basic physics of the problem -observation in different inertial frames of events of two spatially-separated and synchronised clocks. Section 6 contains a brief summary. 

[ . . . . . . . . . . . ] 

6 Summary 
The manifest translational invariance of the TD relations in (2.13) demonstrate the spurious nature of correlated ‚relativity of simultaneity‘ and ‚length contraction‘ effects that have hitherto, following Einstein [1], been derived from the LT, as explained in previous papers by the present author [2, 3, 4] and Section 2 of the present paper. 

Alternative derivations of ‚length contraction‘ from the geometry of the Minkowski plot are shown to be flawed by a sign error in an angle, when drawing the x? and t? axes on the original Minkowski plot [5], which has been propagated, uncorrected, in essentially all text-book treatments of the subject, as well as the use projection operations at variance with those required by the LT. When the correct projection and scaling operations derived in Section 3, directly from the LT, are applied, it is clear, by simple inspection, (see Fig. 3 and Fig. 9b) that there is no ‚length contraction‘ effect.

The ‚relativity of simultaneity‘ effect derived from the Minkowski plot, unlike that obtained in Eqns(2.14) – (2.17), directly from misuse of the LT, is not a direct consequence of the ‚length contraction‘ effect obtained from the same plot. It is, instead, produced by a false identification of certain projections on the plot with the times recorded by two spatially separated and synchronised clocks. It is shown in Fig. 12a, that, even with an incorrect oblique projection procedure, similar to that used to obtain ‚length contraction‘, there is no corresponding ‚relativity of simultaneity‘ effect.

The present author has been able to find only one text book in which the x? and t? axes of the Minkowski plot are correctly drawn [12]. However, the plot is not used in the book in the discussion of the ‚length contraction‘ effect.


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