# History of Topics in Special Relativity/Twin paradox

## Topics in the clock/twin paradox

Date of article creation: 9 November 2023; Last edit: 23 November 2023

a) Who was the first to introduce human beings into the clock/twin paradox?

b) Who was the first to introduce the three clock/brother example that completely removes acceleration from the clock/twin paradox?

• Current historical accounts[S 4][S 5] date it back to Lange (1927)[1] and Lord Halsbury (1957).[2]
• An even earlier discussion of the three clock/brother example is provided in #Wiechert (1920-22).

## a) Human beings

### Einstein (1911)

After w:Albert Einstein in 1905[3] introduced the round-trip clock experiment (i.e. clock/twin paradox),[S 2] he gave a lecture in 16 January 1911 (published 27 November 1911)[4] in which he modified his experiment by using living organisms. He argued that a clock can be seen as a representation of all physical processes, thus by bringing a living organism into a box and then moving it forwards and backwards like the clock before, the journey would only lasted a moment to the moving organism, while at return it would observe that the other remaining organism has already been replaced by new generations a long time ago.

### Lämmel (1911)

w:de:Rudolf Lämmel attended[5] #Einstein's January 1911 lecture and gave a popular report about it and the subsequent discussion with Einstein in the Swiss newspaper "Neue Zürcher Zeitung" published on 28 April 1911, in which he gave further details.[6] Regarding the clock/twin paradox he wrote:

German English translation
Bewegt sich eine Uhr mit Lichtgeschwindigkeit längs einer Geraden, auf der gerichtete Uhren stehen, so scheint die bewegte Uhr, beurteilt vom Standpunkt der ruhenden aus, im oben stizzierten Sinn, stillzustehen. Kehrt die Uhr, nach einem Ruck, mit Lichtgeschwindigkeit wieder zurück zur Zentral-Uhr, so ist, nach Einstein, für den Beobachter bei der Zentral-Uhr die Sache so, als ob ein mit der bewegten Uhr mitgeführter Beobachter (samt dessen Uhr) nicht gealtert hätte. Hinge also des letzteren Alter von den Angaben des ruhenden Beobachters ab, so könnte der von einer großen Reise ins Weltall zurückkehrende Beobachter bei der Zentral-Uhr spätere Generationen antreffen – er selber hätte nicht gealtert. Welche Bedeutung diese ad absurdum geführte Gedankenspielerei etwa hat, läßt sich heute nicht absehen – vielleicht, ja wahrscheinlich ist sie ohne jeden Einfluß auf die tatsächlichen Verhältnisse. Aber man sieht dabei immerhin, daß die Physik imstande ist, die kühnsten Träume der Phantasie noch – zu überbieten. Let a clock be moving at speed of light along a line on which regulated clocks are standing, then the moving clock's hand appears to be (in the sense described above) standing still as judged from the standpoint of the resting one. If the clock, after one jolt, comes back with light speed to the central clock, then according to Einstein the matter presents itself to the observer at the central clock, as if the observer comoving with the clock (together with his clock itself) hasn't been grown older. Thus if the age of the latter would depend on the indications of the resting observer, then the observer returning from a great journey into space could meet later generations at the central-clock – he himself hasn't been grown older. The importance of this play of thought led ad absurdum cannot be seen today – maybe, or even probably, it is without any influence on the actual situations. Though at least one can see that physics is able to – surpass – even the boldest dreams and fantasies.

Lämmel in December 1920 (published 1921)[7] again alluded to Einstein's 1911 lecture, describing a discussion between himself and Einstein. After Einstein concluded that the travelers who came back after their journey will probably meet their former contemporaries as old men while they themselves could have been away for only a few years, Lämmel objected that this conclusion is only drawn with respect to rods and clocks, but not with respect to living beings. Einstein responded though, that all processes in the blood, in the nerves etc. are eventually periodical oscillations, i.e. motions. Yet to any such motion the relativity principle applies, thus the conclusion regarding the unevenly rapid aging it permissive.

While the official publication of Einstein's January lecture mentions the aging of organisms – see #Einstein (1911) – Lämmel recalls the reference to the aging of a human space traveler ("observer returning from a great journey into space" in 1911 or "meet their former contemporaries as old men" in 1921). This means that Einstein was the first to use human beings in the clock/twin paradox in January 16 which was first published by Lämmel in 28 April 1911. In comparison, #Langevin (1911) used space travelers in a lecture on April 10 with publication in July, and #Wiechert (1911) used space travelers in lectures held on March 25 and May 23 with publication in September. It seems very unlikely that before April 28, Lämmel became somehow aware of the content of Langevin's or Wiechert's lectures held a few weeks earlier, in order to use them in his description of Einstein's lecture.

### Langevin (1911)

Independently, w:Paul Langeven (10 April 1911, published July 1911)[8] held a now famous lecture popularizing the clock/twin paradox. See the full Wikisource translation "The Evolution of Space and Time", in which he describes how a traveler leaves Earth in a projectile for 2 years proper time with velocity of 1/20000 less than light speed, but at return he finds out that Earth has aged 200 years. Langevin finds it amusing to realize what the explorer and the Earth would see if they could mutually stay in constant communication during their separation by light signals or by wireless telegraphy, and thus understand how the asymmetry between two measures of the duration of separation is possible.

### Wiechert (1911)

Independently, w:Emil Wiechert (lectures on 25 March and 23 May 1911, submitted July and published September 1911)[9] described the clock/twin paradox with two equal clocks regulated to the same rate and brought to the same pointer position, or the same chemical process shall be introduced two times, or by introducing two life forms that began their life at the same time. At the end of his paper he discussed human travelers whose relative velocity approximating the speed of light by 3 percent, then the ratio of the experienced length of time becomes 4:1. Let's imagine that an observer travels with that velocity in a circular path at a radius of 16 light years through the space of our galaxy, then according to our time calculation he passes by our solar system every 100 years. In his vehicle the centrifugal force will act on him in such a way, that in accordance with the relativity laws it will appear to be equal to the force of gravity acting upon the inhabitants of Earth. Thus the acting forces are only thus big, in order to give our fantasy the possibility to imagine the traveler as a human being. Since we have ${\displaystyle {\sqrt {1-v^{2}/c^{2}}}}$  throughout, proper time flows four times slower for the traveler, than for the inhabitants of the stars. Thus when he comes back to our solar system after 100 of our years, he will feel to have aged only by about 25 years. If he reaches an age of 75 years according to the development of his body and his own time experience, then this corresponds to a triple return to our solar system, i.e. 300 of our Earth years.

Wiechert published another paper in 1915,[10] in which he provided a short historical survey of the clock/twin paradox. He referred to the fact that already Einstein (1905)[3] considered the case of two clocks (“Einstein's clock experiment”), and even though w:Hermann Minkowski himself didn't consider the case, his proper time formula provides the result in a straight forward manner. The latter was done by himself in lectures on 25 March and 23 May 1911, as well as by Langevin published in July 1911. Wiechert pointed out that he himself and Langevin used “humorist” examples in order to clarify the situation: While Wiechert argued that one has to make a journey in order to stay young, Langevin argued that one has to romp about in a laboratory in order to stay young. Both of them used human beings, arguing that their physical and mental life should have been influenced in the same way as any other process in nature.

The dates given by Wiechert in 1915 are not complete. The correct ones are:

• Langevin's lecture on 10 April 1911, published in July.
• Wiechert's lectures on 25 March and 23 May 1911, submitted on July 26, published in September.
• He was still unaware of Einstein's lecture from January 1911, published in November.

### Müller (1911)

The freelance writer and law student Fritz Müller (who was later known as w:de:Fritz Müller-Partenkirchen) attended[5] #Einstein's January 1911 lecture and wrote – similar to #Lämmel (1911) before – a popular report about it and the subsequent discussion with Einstein in the German newspaper "Berliner Tageblatt" published in two parts in 16. and 23. October 1911, in which he gave further details.[11] Regarding the clock/twin paradox he wrote:

German English translation
Zwei gleichgehende Uhren sollen je einen Beobachter haben und nebeneinander ruhen. Nun soll die eine mit ihrem Beobachter plötzlich mit Lichtgeschwindigkeit in den Weltenraum hinausreisen. Vorher haben die beiden vereinbart, sich alle Sekunden mit einem Lichtsignal die Zeit zu telegraphieren. [...] In unserem Grenzfall, wo die Reise mit Lichtgeschwindigkeit vor sich geht, müßte der ruhende Beobachter erklären, jene andere Uhr käme in der Zeit überhaupt nicht voran. Die Zeit stünde dort still. Tatsächlich kommen die Einsteinschen Gleichungen zu diesem Resultat. Für den mit der Uhr reisenden Beobachter, sagt Einstein, gelte dasselbe. Das heißt, im Urteil des Zurückbleibenden würde jener niemals alt. „Und wenn er auf einer gebrochenen Reiselinie wieder an seinen Ausgangspunkt zurückkehrte?" fragt man den Vortragenden in der Diskussion. – „So bliebe er in unserem Urteil so jung wie bei der Ausreise," erwidert Einstein mit vollem Ernst, „selbst wenn wir Zurückgebliebenen inzwischen Männer mit weißen Bärten geworden sind – die Gleichungen liefern für jede Richtung der Bewegung, auch für eine gebrochene Bewegung, unerschütterlich die selben Resultate." – Wir sehen einander an. Das klingt märchenhaft. Märchenhaft? Gewiß, die alten Märchen vom Mönch von Heisterbach, vom Rip van Winkle, von Urashima Taro steigen auf. Merkwürdig, wie die Volksphantasie bei den Deutschen, bei den Amerikanern, bei den Japanern in der gleichen Richtung gearbeitet hat – alle drei Märchen erzählen ja von Leuten, deren Leben still steht, viele hundert Jahre lang, während die andern altern. So fanden sie bei ihrer Rückkehr ein anderes Land und eine andere Generation. Two synchronous clocks at rest next to each other, shall each be accompanied by an observer. Now one of them, together with its observer, suddenly travels into space at the speed of light. Previously, both have arranged that every second they telegraph their time to each other using light signals. [...] In our limiting case where the journey happens at light speed, the resting observer would have to declare that the other clock would not proceed in time at all. Time would stand still at this place. Einstein's equations indeed produce this result. As to the observer traveling with the clock, says Einstein, the same is true. That means in the judgement of the remaining one, the other one would never become old. Then the lecturer [i.e. Einstein] was asked in the discussion: "And if he comes back to his starting point on a curved travel path?", to which Einstein replied in full earnest: "Then in our judgement he would remain as young as he was at departure, even if we remaining ones became men with white beards in the meantime, the equations unshakably give the same result in every direction of motion, also for curved motion". We look at each other. That sounds fabulous. Fabulous? Of course, the old fairy tales of w:The monk of Heisterbach or w:Rip Van Winkle or w:Urashima Tarō come forward. Strange, how the folk fantasy of the Germans, the Americans, the Japanese worked in the same direction, all three fairy tales indeed tell about people whose life stands still, many hundred years long, while the other ones grow old. Thus they found another country and another generation when they returned.
Müller's account confirms #Lämmel (1911) that Einstein indeed mentioned human beings, but his description also suggests that Einstein was the first to use mutually sent light signals. However, as this was published in October, it cannot be excluded that Müller's description of light signals was influenced by #Langevin (1911) (published in July) in which light signals were used as well.

### Weyl (1918)

w:Hermann Weyl (March 1918)[12] argued that the life process of a man can very well be compared with a clock. Of two twin brothers (German: Zwillingsbrüder) separated at a worldpoint A, one remains at home while the other undertakes travels at velocities close to the speed of light. Then the traveling one will be noticeably younger than the remaining one at re-encounter.

## b) Three clock/brother example

### Wiechert (1920-22)

Wiechert (December 1920, published 1921) showed how to remove all accelerations from the clock/twin paradox.[13] There are three bodies A, B, C, which move undisturbed (inertially) in different directions. A and B pass by each other at time (1), B and C pass by each other at a later time (2), and C and A finally pass by each other at time (3). Thus the condition of C is so to speak the continuation of the condition of B. For instance, on any of the three bodies one can count the oscillations of light of a certain spectral-line, then the application of time dilation shows that the combined sum of all oscillations on B and C is smaller than the number of oscillations on A alone. This difference can be made arbitrarily large when the speed of B and C is brought arbitrarily close to the speed of light. For instance, one can imagine that on B and C together only 1 oscillation happened, while trillions of oscillations happened on A. Wiechert also held that one can replace the light oscillations by the life functions of human-like beings which live on A, B and C. For instance, while the inhabitants of B and C only had time for one meal, there were arbitrarily many generations at A who follow after each other by death and birth.

Wiechert (September 1921, published 1922)[14] extended his previous variant of the clock/twin paradox without acceleration by arbitrarily increasing the number of bodies. So all participating bodies ${\displaystyle A}$ , ${\displaystyle B_{1}}$ , ${\displaystyle B_{2}}$ , ... remain unaccelerated, with the B's being the sequential members of a relay race (German: Stafette) in which any B continues the fate of the previous B, all having same speed v in different directions relative to A. The "coincidence" ${\displaystyle I}$  is the event when A coincides with ${\displaystyle B_{1}}$ , while coincidence ${\displaystyle II}$  is the event when A finally coincides with the last of the B's. So from A's perspective, the Lorentz transformation shows that the processes on the relay bodies are slowed down in the ratio ${\displaystyle 1:{\sqrt {1-v^{2}/c^{2}}}}$ . For instance, if one considers light of a certain spectral line emitted from sources on the bodies ${\displaystyle A}$ , ${\displaystyle B_{1}}$ , ${\displaystyle B_{2}}$ , ..., it follows that there should be fewer oscillations between coincidence ${\displaystyle I}$  and ${\displaystyle II}$  on the relay sequence (consisting of the B's) than on A alone, with the counting performed on the respective bodies themselves. As is known, instead of light oscillations one can also choose the aging of life forms.

Wiechert's thought experiments are technically correct descriptions of the clock/brother example. However, Wiechert promoted the non-standard interpretation of special relativity according to which speeds and directions are absolute with respect to an aether.

## References

### Historical references

1. Lange (1927), p. 25
2. Halsbury (1957), p. 178
3. Einstein (1905), p. 904f
4. Einstein (1911), p. 10.
5. Einstein, Müller, Lämmel (1911/12)
6. Lämmel (1911), p. 1
7. Lämmel (1920/21), p. 84ff
8. Langevin (1911), p. 48ff;
9. Wiechert (1911), p. 745f.; 757f.
10. Wiechert (1914/15), p. 46
11. Müller (1911), second part published 23.10.1911
12. Weyl (1918), p. 147f.
13. Wiechert (1920), p. 46f
14. Wiechert (1921/22), p. 25ff

### Secondary sources

1. Miller (1981), section 7.4.13 with reference to Einstein, Langevin, Wiechert, Laue
2. Pesic (2003), section 2 with reference to Einstein, Langevin, Laue
3. During (2014), p. 520f with reference to Einstein, Langevin, Laue, Weyl
4. Debs & Redhead (1996), p. 385 with reference to Halsbury
5. Alizzi et al. (2022), Appendix B with reference to Lange and Halsbury
• Alizzi, A., Sen, A., & Silagadze, Z. K. (2022), "Do moving clocks slow down?", European Journal of Physics, 43 (6): 065601, arXiv:2209.12654, doi:10.1088/1361-6404/ac93ca{{citation}}: CS1 maint: multiple names: authors list (link)
• Debs, T. A., & Redhead, M. L. (1996), "The twin paradox and the conventionality of simultaneity", American Journal of Physics, 64 (1): 384–392, doi:10.1119/1.18252{{citation}}: CS1 maint: multiple names: authors list (link)
• During, É. (2014), "Langevin ou le paradoxe introuvable", Revue de métaphysique et de morale, 84: 513–527, doi:10.3917/rmm.144.0513
• Miller, A. I. (1981), Albert Einstein's special theory of relativity. Emergence (1905) and early interpretation (1905–1911), Reading: Addison–Wesley, ISBN 978-0-201-04679-3
• Pesic, P. (2003), "Einstein and the twin paradox", European Journal of Physics, 24 (6): 585–590, doi:10.1088/0143-0807/24/6/004