Inaccuracy That Makes Learning Difficult to Study the Article ”On the Electrodynamics of Moving Bodies”

1. Introduction

A. Einstein’s work “On the Electrodynamics of Moving Bodies” [1], [p.7-35] [5,6], [p.140-171], published in 1905, outlining the main provisions of the Special Theory of Relativity, is now deservedly considered classic. It is all the more important to detect and exclude typos and inaccuracies from it. Typos and inaccuracies can be purely stylistic, spelling, factual (dates, proper names, etc.) [4] and semantic [3]. The first three categories, of course, are not of great interest, although they are found both in reprints published during the author’s lifetime and in subsequent ones, but correcting them is only a technical task. Of greatest interest are semantic typos concerning inaccuracies in formulas, descriptions of experimental results, etc., since, in addition to correcting them, when discussing them, whole discussions arise that make it possible to determine whether the inaccuracy actually occurs, or whether the author’s intention is much deeper than the first glance. In addition, undetected or uncorrected inaccuracies can become a source of difficulties when teaching physics, in particular, the foundations of the Special Theory of Relativity based on primary sources, and also make it difficult to understand the author’s logic when studying them independently.

A preliminary search on the Internet, primarily in Russian-language sources, allowed us to find only one known published semantic typo concerning the incorrect indication of the speed at which the frequency, in the formula for the relativistic Doppler effect, reaches infinity [3]. Unfortunately, it is not enough to simply point out a typo, since, despite the passage of more than half a century, the above typo continues to appear in reprints of Einstein’s work [1], [p.26] [6], [p.161].

2. Context Description

A. Einstein introduces the following procedure (criterion) to determine the synchronism of clocks A and B, located at different points in space A and B:

• “For, suppose a ray of light leaves from A toward B at "A-time" $t_A$, is reflected from B toward A at "B-time"  $t_B$, and arrives back at A at "A-time" ${t\prime }_A$. The two clocks are synchronous by definition if  $t_B-t_A={t\prime }_A-t_B″$ [1], [p.9] [6], [p.142].

And further, A. Einstein describes the result of a thought experiment demonstrating the relativity of simultaneity:

We further imagine that each clock has an observer co-moving with it, and that these observers apply to the two clocks the criterion for synchronism formulated in §1. Suppose a ray of light starts out from A at time ${\mathit{t}}_{\mathit{A}}$, is reflected from B at time ${\mathit{t}}_{\mathit{B}}$, and arrives back at A at time ${t\prime }_A$. Taking into account the principle of the constancy of the velocity of light, we find that

$$\begin{array}{l}{\mathit{t}}_{\mathit{B}}-{\mathit{t}}_{\mathit{A}}=\frac{{\mathit{r}}_{\mathit{A}\mathit{B}}}{(\mathit{v}-\mathit{v})}\hspace{1em}and\hspace{1em}{\mathit{t}\prime }_{\mathit{A}}-{\mathit{t}}_{\mathit{B}}=\frac{{\mathit{r}}_{\mathit{A}\mathit{B}}}{(\mathit{v}+\mathit{v})}\hspace{1em}\hspace{1em}\hspace{1em}\left(\right(1),{\mathit{t}\prime }_{\mathit{A}}-{\mathit{t}}_{\mathit{B}}!={\mathit{t}}_{\mathit{B}}-{\mathit{t}}_{\mathit{A}}-\\ (D.K.)\end{array}$$

where $r_{AB}$ denotes the length of the moving rod, measured in the system at rest. The observers co-moving with the moving rod would thus find that the two clocks do not run synchronously while the observers in the system at rest would declare them synchronous.” [1], [p.12-13, 6, p.145]

Figure 1. Explanation of the result of a thought experiment demonstrating the relativity of simultaneity: a) fulfillment of Einstein’s synchronicity criterion for observers on a rod b) the same for observers at rest.

Figure 1. Explanation of the result of a thought experiment demonstrating the relativity of simultaneity: a) fulfillment of Einstein’s synchronicity criterion for observers on a rod b) the same for observers at rest.

In our opinion, in the above quote there is a semantic inaccuracy; the preposition “not” is placed in the wrong place where it should be in meaning. The sentence in the quotation should read like this: “The observers co-moving with the moving rod would thus find that the two clocks do run synchronously while the observers in the system at rest would declare them not synchronous.”

3. Rationale and Proof

To substantiate and prove the above, one should turn to the meaning of formulas (1), which determine how much time it takes for a ray of light to travel from the clock A installed at one end of the moving rod to the clock B installed at the other end of this rod and back.

If these times are equal, the clocks are considered to be running synchronously, that is, showing the same time, or having the same clock hands. Since the denominators of expressions (1) use the value of the rod speed v relative to a stationary reference frame, this speed cannot be determined by observers on the rod; for them, the time of passage of the rays from clocks A to B and back must always be equal, regardless of whether the observers move with the rod or the rod is at rest.

If it were possible for observers on the rod, by conducting one or another experiment, to determine whether the rod is moving translationally and uniformly, or at rest, this would be a violation of the 1st postulate of the Special Theory of Relativity, according to which there is no experiment that would show the difference between a system at rest and moving uniformly and rectilinearly. In other words, for observers on the rod, formulas (1) will always look like:

$${\mathit{t}}_{\mathit{B}}-{\mathit{t}}_{\mathit{A}}=\frac{{\mathit{r}}_{\mathit{A}\mathit{B}}}{\left(\mathit{I}\mathit{N}\right)}\hspace{1em}{\mathit{t}\prime }_{\mathit{A}}-{\mathit{t}}_{\mathit{B}}=\frac{{\mathit{r}}_{\mathit{A}\mathit{B}}}{\left(\mathit{I}\mathit{N}\right)}\hspace{1em}{\mathit{t}}_{\mathit{B}}-{\mathit{t}}_{\mathit{A}}={\mathit{t}\prime }_{\mathit{A}}-{\mathit{t}}_{\mathit{B}}$$

(2)

Consequently, only observers in a fixed, at rest coordinate system can calculate time using formulas (1).

It is they, and only they, who will be able to see how a beam of light from a source in clock A “catch up” with those moving away from it at the speed of movement of the rod v clock B, and will be able to notice the inequality in the time of the reverse movement of the beam from clock B to clock A, since in the latter case, clock A moves towards the ray of light released from clock B.

And, having clarified the inequality of these times, only observers from a system at rest will be able to draw a conclusion about the non-synchronism of the clock on the moving rod.

In the lectures on physics by R. Feynman [7], a description of a similar thought experiment is given, slightly different in technical details (instead of comparing the time it takes for the forward and backward rays to travel the length of the rod, Feynman examines a spacecraft with a flash of light emanating from its center, synchronizing the clocks located at the stern and bow, respectively, and shows that for an astronaut (observer on the rod), the clocks will run synchronously, but for an observer from outside the ship (from a system at rest) - asynchronously, since the time it takes for the beam to travel to the bow of the ship will be longer, than to the stern, since the bow, for an observer at rest, “runs away” from the beam, and the stern moves towards the beam. Thus, in the description of Einstein’s thought experiment as presented by R. Feynman, the inaccuracy has been corrected: “So it turns out that when the cosmonaut believes that events in two places on the ship occurred simultaneously(for the same value of t' in its coordinate system), then in another coordinate system, the same t’ corresponds to different t values!(selections made by the author of the article - D.K.)” [7, p.18, my emphasis - D.K.] [8], [p.15-6]

4. Objections

In our opinion, the presence of inaccuracy is quite obvious, however, we can briefly touch on possible objections [2]:

4.1. The clock on the rod is set by the clock in a stationary, resting frame of reference, so it always shows the time of the frame at rest, and since the rod moves relative to this frame, observers on the rod will see the non-synchronism of their clocks, just as observers in a frame at rest would see the non-synchronism the progress of the clock on the rod.

However, when illustrating the relativity of simultaneity, Einstein does not touch upon the question of what exactly the hands of a clock show - especially since the hands of all clocks, both those located on a rod and those at rest, always show the same thing - this is directly stated in the article: “...to both ends of the rod (A and IN) a clock is attached that is synchronous with the clock of the system at rest, that is, its readings correspond to the “time of the system at rest” in those places in which these clocks are located; therefore, these clocks are “synchronous in a system at rest” [1], [p.12].

And we are talking only about whether the criterion of clock synchronicity is fulfilled according to the procedure proposed by Einstein during the movement of the rod, and the obligatory nature of its fulfillment for observers on the rod clearly follows from the firstpostulate of the Special Theory of Relativity..

4.2. Since the rod is moving, time slows down for observers on the rod, and the clock on the rod is set to the time of the system at rest, so observers on the rod can see that the clocks are not synchronous.

This does not correspond to the thought experiment proposed in the article, since to illustrate the relativity of simultaneity, A. Einstein does not use any concepts specific to the special theory of relativity (does not consider the dilation of time on a moving rod, the shortening of its length for stationary observers, etc.), the situation is considered in ordinary Euclidean space [1], [p.8] [6], [p.141].

4.3. In Feynman's description, the moving clocks are synchronized "according to Einstein." Therefore, in each place they fly past, they do not give the same readings as the readings of stationary clocks installed in the same places, synchronized “according to Einstein” in their stationary state.

And in Einstein's description there is a moving clock Not synchronized according to Einstein. Einstein specially synchronized the moving clocks in that description differently (he synchronized them to the same readings for all clocks, just as it was always assumed in non-relativistic physics) - they give in every place they fly past exactly the same readings as the readings stationary clocks installed in the same places, synchronized “according to Einstein” in their stationary state.

For Einstein, synchronous clocks are those clocks that are synchronized according to Einstein, that is, if the time of passage of a ray of light between the clocks and back is equal, the clocks are synchronous, otherwise they are asynchronous. It doesn’t matter what time the clock hands show, their synchronicity is not determined by the hands. And on the rod, always and in any circumstances, performing clock synchronization according to Einstein (measuring the time of passage of a light beam between one to the other and back) will always give a positive result. In the description of the thought experiment in the article, A. Einstein specifically indicated that all clocks are set according to the clock of the system at rest - therefore, all time stamps are made in the system at rest.

Let us note once again that synchronization according to Einstein is not about setting the clock hands, but about comparing the time it takes for a beam of light to travel from one to the other and back. This is quite obvious from Einstein's description: The two clocks are synchronous by definition if $t_B-t_A={t\prime }_A-t_B″$ [1], p.9] [6], [p.142]

Feynman's scientific experiment uses exactly the same clock synchronization mechanism.

5. Conclusions

In our opinion, the presence of inaccuracy proved above is quite obvious. The more interesting question is why, for more than 100 years since the publication of A. Einstein’s seminal work, no one has yet paid attention to this inaccuracy and proposed to correct it, perhaps saving students and students studying primary sources from headaches and the impossibility of constructing a logically consistent presentation of one of the fundamental principles of the special theory of relativity.

As said over 50 years ago,“It seems appropriate with this letter to draw the attention of readers and, in particular, editors to the fact that reverence for the classics of science should not be extended to obvious typos made in the first publications of their works.” [3].

We join this statement.