History of Topics in Special Relativity/Twin paradox

History of Topics in Special Relativity () Multiple chapter topics History of Lorentz transformations History of relativistic four vectors History of relativistic tensors Single page topics History of twin paradox History of Penrose–Terrell effect History of time dilation History of three-acceleration transformation Müller's history of relativity

Early history of the twin paradox

Date of article creation: 9 November 2023; Last major revision: 2 March 2026

a) When was the w:twin paradox applied to life forms and human beings?

• Historical accounts report that § Einstein 1911-HU discussed the aging of living organisms, and that § Langevin 1911-HU and § Wiechert 1911-HU explicitly discussed the aging of human beings.

• More details in sections §§ Human beings in 1911​ and Twins from 1911 to 1920, including newspaper articles from 1911 written by § Lämmel 1911-HU and § Müller 1911-HU that clearly show that Einstein was the first to explicitly discuss the aging of human beings as well.

b) Who was the first to formulate the principle of maximal proper time along straight worldlines, upon which differential aging in the standard twin paradox is based?

• Historical accounts mention Langevin (1911), Laue (1911).

• More details in section § Maximal proper time with the contributions of Langevin (1911), Wiechert (1911), Study (1911), Laue (1911-13).

c) Who was the first to formulate inverse triangle inequality in Minkowski space, which represents the simplest version of the twin paradox?

• See details in section § Triangle inequality with the contributions of Robb (1914-20), Eddington (1922), Rogers (1922).

d) Who was the first to show that any influence of proper acceleration on clocks can be neglected in the computation of the twin paradox from the viewpoint of the stay-at-home twin?

• Historical accounts mention Einstein (1911), Laue (1913).

• More details in section § Negligibility of proper acceleration with the contributions of Einstein (1911), Wiechert (1911), Laue (1913), Lorentz (1913).

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

• Historical accounts date it back to Lange (1927) and Lord Halsbury (1957).

• More details in section § Relay (three brothers) experiment with the contributions of Grünbaum (1911) and Wiechert (1920-22).

f) Who was the first to use acceleration as an asymmetry indicator?

• Historical accounts mention Langevin (1911), Einstein (1918).

• More details in section § Acceleration as asymmetry indicator with the contributions of Langevin (1911), Sommerfeld (1913), Lorentz (1913), Einstein (1914-20).

g) Who was the first to use different frame distribution as asymmetry indicator as an asymmetry indicator?

• Historical accounts mention Laue (1911-13).

• More details in section § Frame distribution as asymmetry indicator with the contributions of Laue (1911-13), Bloch (1918).

h) Who was the first to describe the perspective of the traveler?

• Historical accounts mention Langevin (1911), Lorentz (1914), Einstein (1918).

• More details in section § Perspective of the traveler with the contributions of Langevin (1911), Lorentz (1913-14), Einstein (1918), Thirring (1921).

i) Who was the first to describe a round-trip experiment in curved spacetime?

• See section § Curved spacetime with the contribution of Becquerel (1922).

j) Who was the first to denote the round-trip experiment as paradoxical?

• Historical accounts point to Laue (1911).

• See section § Paradoxical? for details.

k) Who was the first to misunderstand the twin paradox?

• See section § Misunderstandings with the contributions of Berg (1910), Wiechert (1911), Campbell (1911/12), Gruner (1912).

Human beings in 1911

Einstein
In 1905 he showed that a clock moving on a round-trip away from A and back along a polygonal or curved path, is retarded with respect to a clock stationary at A by approximately ${\tfrac {1}{2}}t(v/V)^{2}$ at reunion. For example, a clock on the equator is retarded with respect to a clock on the pole. He described this consequence as being "peculiar" (German: eigentümlich). In a lecture given on January 1911 (published in November), he extended this "funny" (German: drollig) experiment to living organisms: Einstein wrote English translation Wenn wir z. B. einen lebenden Organismus in eine Schachtel hineinbrächten und ihn dieselbe Hin- und Herbewegung ausführen lassen wie vorher die Uhr, so könnte man es erreichen, dass dieser Organismus nach einem beliebig langen Fluge beliebig wenig geändert wieder an seinen ursprünglichen Ort zurückkehrt, während ganz entsprechend beschaffene Organismen, welche an den ursprünglichen Orten ruhend geblieben sind, bereits längst neuen Generationen Platz gemacht haben. Für den bewegten Organismus war die lange Zeit der Reise nur ein Augenblick, falls die Bewegung annähernd mit Lichtgeschwindigkeit erfolgte! For example, if we put a living organism in a box and make it undergo the same back and forth movement as the clock before, we could achieve that this organism returns to its original location with arbitrary little change after a flight of arbitrary length, whereas completely identical organisms that remained at rest in the original location have long since made room for new generations. To the moving organism, the long journey was only a moment if the movement happened close to the speed of light! Two participants of that lecture, § Lämmel 1911-HU and § Müller 1911-HU, report that Einstein also talked about the aging of human beings.
Lämmel
He attended Einstein's 1911 lecture and gave a popular report about it in the Swiss newspaper "Neue Zürcher Zeitung" published on 28 April 1911, including additional details. Regarding the round-trip clock experiment he wrote: Lämmel wrote 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 standing still (in the sense described above) 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, 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) again alluded to Einstein's lectures in Zürich (possibly the one from 1911, and maybe also later ones), 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 (§ Einstein 1911-HU) mentions the aging of organisms, Lämmel recalls the reference to the aging of a human space traveler ("observer returning from a great journey into space"). This means that Einstein was the first to use human beings in the clock/twin paradox on January 16 which was first published by Lämmel on April 28, 1911. In comparison, § Langevin 1911-HU used space travelers in a lecture on April 10 with publication in July, and § Wiechert 1911-HU used space travelers in lectures held between March 25 and May 23 with publication in July/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
On 10 April 1911, published July 1911, he held a now famous lecture popularizing the clock/twin paradox which he derived from the proper time integral as described in § Langevin 1911-PT. He demonstrated that a moving radioactive sample of radium is less evolved and less aged and therefore more active at return then the ones that remained in the laboratory. He also used light signals and the Doppler effect to visualize the effect. The most famous part concerned his description of the aging of human space travelers: Langevin wrote English Wikisource translation Cette remarque fournit le moyen, à celui d’entre nous qui voudrait y consacrer deux années de sa vie, de savoir ce que sera la Terre dans deux cents ans, d’explorer l’avenir de la Terre en faisant dans la vie de celle-ci un saut en avant qui pour elle durera deux siècles et pour lui durera deux ans, mais ceci sans espoir de retour, sans possibilité de venir nous informer du résultat de son voyage puisque toute tentative du même genre ne pourrait que le transporter de plus en plus avant. Il suffirait pour cela que notre voyageur consente à s’enfermer dans un projectile que la Terre lancerait avec une vitesse suffisamment voisine de celle de la lumière, quoique inférieure, ce qui est physiquement possible, en s’arrangeant pour qu’une rencontre, avec une étoile par exemple, se produise au bout d’une année de la vie du voyageur et le renvoie vers la Terre avec la même vitesse. Revenu à la Terre ayant vieilli de deux ans, il sortira de son arche et trouvera notre globe vieilli de deux cents ans si sa vitesse est restée dans l’intervalle inférieure d’un vingt-millième seulement à la vitesse de la lumière. Les faits expérimentaux les plus sûrement établis de la physique nous permettent d’affirmer qu’il en serait bien ainsi. This remark provides the means for any among us who wants to devote two years of his life, to find out what the Earth will be in two hundred years, and to explore the future of the Earth, by making in his life a jump ahead that will last two centuries for Earth and for him it will last two years, but without hope of return, without possibility of coming to inform us of the result of his voyage, since any attempt of the same kind could only transport him increasingly further. For this it is sufficient that our traveler consents to be locked in a projectile that would be launched from Earth with a velocity sufficiently close to that of light but lower, which is physically possible, while arranging an encounter with, for example, a star that happens after one year of the traveler's life, and which sends him back to Earth with the same velocity. Returned to Earth he has aged two years, then he leaves his ark and finds our world two hundred years older, if his velocity remained in the range of only one twenty-thousandth less than the velocity of light. The most established experimental facts of physics allow us to assert that this would actually be so. Reading his lecture in full, one finds the word "paradoxical" only in relation to the constancy of light speed, not on relation to the round-trip clock experiment.
Wiechert
In lectures on 25 March and 23 May 1911, submitted July and published September 1911, he described the round-trip clock experiment with two equal clocks regulated to the same rate and brought to the same pointer position, or by introducing the same chemical process two times, or by introducing two life forms that began their life at the same time. At the end of his paper he applied this to human travelers: Wiechert wrote English translation Nehmen wir aber wieder eine Relativgeschwindigkeit an, die bis auf 3 Proz. der Lichtgeschwindigkeit nahekommt, dann wird das Verhältnis der empfundenen Zeitlängen wie 4:1. Das Bild mag etwas weiter noch ausgemalt werden. Denken wir uns, daß ein Beobachter durch den Raum unseres Sternhimmels mit dieser Geschwindigkeit in einer Kreisbahn mit einem Radius von 16 Lichtjahren fährt, dann wird er nach unserer Zeitrechnung nach je 100 Jahren wieder an unserem Sonnensystem vorüberkommen. In seinem Gefährt wird dabei die Zentrifugalkraft so auf ihn einwirken, daß sie gemäß den Relativitätsgesetzen der Einwirkung der Schwerkraft auf uns Erdenbewohner gleich erscheint. Es sind also die wirkenden Kräfte nur so groß, daß der Phantasie die Möglichkeit geboten wird, den Reisenden als menschliches Wesen zu denken. Da hier dauernd ${\sqrt {1-v^{2}/c^{2}}}$ ist, fließt die Eigenzeit für den Reisenden viermal langsamer dahin, als für die Bewohner der Gestirne. Wenn er also nach 100 unserer Jahre wieder zu unserem Sonnensystem zurückkehrt, wird er sich selbst nur um 25 Jahre gealtert fühlen. Erreicht er nach der Entwicklung seines Körpers und nach seiner Zeitempfindung ein Alter von 75 Jahren, so entspricht dies doch einer dreimaligen Wiederkehr zu unserem Sonnensystem, also 300 unserer Erdenjahre. Yet if we again assume a relative velocity approximating the speed of light by 3 percent, then the ratio of the experienced duration of time becomes 4:1. This image can be further extended. Let's imagine that an observer travels with that velocity on 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 ${\sqrt {1-v^{2}/c^{2}}}$ 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 threefold return to our solar system, i.e. 300 of our Earth years. a) Wiechert (1915) later provided a short historical survey of the clock/twin paradox. He referred to the fact that already § Einstein 1905 considered the case of two clocks ("Einstein's clock experiment"), and even though 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. b) The dates given by Wiechert (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 1911.
Müller
The freelance writer and law student Fritz Müller (who was later known as Müller-Partenkirchen) attended Einstein's lecture and wrote a popular report about it in the German newspaper "Berliner Tageblatt" on 16th and 23rd October 1911, in which he gave further details (compare with § Lämmel 1911-HU). Regarding the clock/twin paradox he wrote: Müller wrote 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 judgment 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 judgment 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-HU 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-HU, published in July, in which light signals were used as well.

Twins from 1911 to 1920

We now provide a list of authors who employed twins, i.e. two life forms or humans that initially were of same age when the round-trip began:

Author	Date	Description
Wiechert	1911	Two life forms that begin their life at the same time (German: "Zwei Lebewesen [..] die ihr Leben gleichzeitig beginnen"), of which the moving one returns retarded in its progression with respect to the stationary one.
Gruner	1912	Two persons of same age (French: "deux personnes du même âge"), of which the moving one returns less developed than stationary one.
Laue	1913	The moving life form returns younger than its former agemates (German: "ehemaligen Altersgenossen").
Weyl	Easter 1918	German original English translation Von zwei Zwillingsbrüdern, die sich in einem Weltpunkt A trennen, bleibe der eine in der Heimat (d. h. ruhe dauernd in einem tauglichen Bezugsraum), der andere aber unternehme Reisen, bei denen er Geschwindigkeiten (relativ zur »Heimat«) entwickelt, die der Lichtgeschwindigkeit nahekommen; dann wird sich der Reisende, wenn er dereinst in die Heimat zurückkehrt, als merklich jünger herausstellen denn der Seßhafte. Suppose we have two twin-brothers who take leave from one another at a world-point A, and suppose one remains at home (that is, permanently at rest in an allowable reference-space), whilst the other sets out on voyages, during which he moves with velocities (relative to “home”) that approximate to that of light. When the wanderer returns home in later years he will appear appreciably younger than the one who stayed at home. Weyl was the first to explicitly use twins in relation to the round-trip experiment. The fourth edition (1920) of that book was translated from German into English and French in 1922.
Einstein	1920/21	German original English translation Trifft A wieder bei B ein, so kann es sich ereignen, daß der beharrende Zwilling inzwischen 60 Erdjahre alt geworden ist, während der zurückkehrende nur 15 Jahre zählt, oder sich gar noch im Säuglingsstadium befindet. [..] Bei diesen Zwillingen, erklärte Einstein, haben wir zunächst eine Gefühls -Paradoxie vor uns. Eine Denk-Paradoxie würde indeß nur dann vorliegen, wenn sich für das Verhalten der beiden Geschöpfe kein zureichender Grund anführen ließe. If A then returns to B, it may happen that the twin who stayed at home is now sixty years old, whereas the wanderer is only fifteen years of age, or is perhaps only an infant still. [..] In the case of these two twins, Einstein declared, we have merely a paradox of feeling. It would be a paradox of thought only if no sufficient ground could be suggested for the behaviour of these two creatures. This was based on an interview of Einstein by Moszkowski. While the expression "clock paradox" was used since 1911/12 (see section § Paradoxical?), this seems to be the first time that it was rebranded as "twin paradox". The copyright mark indicates 1920, while the title page indicates 1921. The translation from German into English also appeared in 1921.

Maximal proper time

Author	Early examples
Langevin 1911	In April 1911 (published July), he described the round-trip experiment without formulas using two portions of matter present at two events happening at the same place. The integration of proper time along the entire wordlines shows that the portion of matter that starts a closed cycle by receding and finally coming back, will have a smaller proper time than the one that stayed behind. In October 1911 (published 1912), Langevin again showed that the portion of matter that described a closed cycle will have a smaller proper time $R$ than the one that stayed in an inertial frame, which is defined by the equation: ${\begin{matrix}V^{2}\left(t-t_{0}\right)^{2}=d^{2}-R\\\left[d^{2}=\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}+\left(z-z_{0}\right)^{2}\right]\end{matrix}}$
Wiechert Lectures March-May 1911 submitted July published September	Let two equal processes be observed in two equal material systems colocated in two moments (1) and (2), and let there velocities have been changed in arbitrarily different ways in the meantime. It follows that the ratio of advancement of those processes is given by the two intervals $\Delta \tau$ of their respective proper times. He concluded that any round-trip clock experiment can be easily comprehended from that theorem by computation. The corresponding integral is: $\Delta \tau =\int _{1}^{2}d\tau =\int _{1}^{2}dt{\sqrt {1-{\frac {{\mathfrak {v}}^{2}}{c^{2}}}}}$
Study June 1911	Minkowski's concept of worldlines implies that the straight path between two points of the same worldline is the longest among all paths between those points, if the path length on a worldline is defined by the related proper time. Study's book was purely mathematical without mentioning clocks or the round-trip experiment, alluding to his result only in a footnote.
Laue 1911-13	In December 1911 (published 1912), Laue showed without formulas that the round-trip experiment is represented by a curved worldline, which at worldpoint A decomposes into a row of curves, after which all of them will be re-united at worldpoint B to a single line. Of all curves connecting the points A and B having time-like direction throughout, the straight connection has the longest proper time. In December 1912 (published 1913) in the second edition of this relativity book, Laue described the proper time integral between events 1 and 2 of a slowly accelerated clock covering a broken line and a stationary clock covering a straight worldline. Of all worldlines covering 1 and 2, the straight line has the longest proper time. Therefore the traveling clock in the round-trip experiment is retarded at reunion, because its curved worldline corresponds to a shorter proper time. This result he presented in terms of the following inequality, of which the right-hand side refers to the straight curve of the stationary clock, while all others possible curves are represented on left-hand side: ${\tfrac {1}{c}}\int _{1}^{2}{\sqrt {du^{2}-\left(dx^{2}+dy^{2}+dz^{2}\right)}}<{\tfrac {1}{c}}\int _{1}^{2}du$ Similar treatments can be found in the textbooks of Sommerfeld (1913), Weyl (1918), Pauli (1921), Kopff (1921), Becquerel (1922).

Triangle inequality

Author	Early examples
Robb 1914-1920	In 1914 he showed that there are three types of triangles formed by intervals in Minkowski space, depending on whether one deals with "separation lines" (spacelike intervals), "optical lines" (lightlike intervals), or "inertia lines" (timelike intervals representing the path of nonaccelerated particles defined by ${\scriptstyle \left(x_{1}-x_{0}\right)^{2}+\left(y_{1}-y_{0}\right)^{2}+\left(z_{1}-z_{0}\right)^{2}-c^{2}\left(t_{1}-t_{0}\right)^{2}<0}$). As to a triangle formed by inertia lines, he showed that the sum of a certain two sides is less than that of the third one. So the triangle inequality derived from time-like intervals in Minkowski space is inverse to the inequality in Euclidean space. This inverse inequality directly represents the most simple variant of the twin paradox: the traveler follows two sides of the time-triangle, while the stay-at-home observer follows the third side indicating maximal proper time. In 1920 Robb gave a numerical example of the triangle ABC with time-like intervals ("inertia lines") defined by coordinates ${\begin{matrix}&x&y&z&t\\A\ &0&0&0&0\\B\ &0&0&0&10\\C\ &4&0&0&5\end{matrix}}$ which he plugged into ${\bar {s}}^{2}=\left(t_{1}-t_{0}\right)^{2}-\left(x_{1}-x_{0}\right)^{2}-\left(y_{1}-y_{0}\right)^{2}-\left(z_{1}-z_{0}\right)^{2}$ from which he obtained the sides AB=10, AC=3, CB=3 and the inequality $AC+CB<AB$.
Eddington 1922	He distinguished between the "space-triangle" for spacelike intervals, and the "time-triangle" for time-like intervals. The latter is measured with a clock from A to B and from B to C, with the sum of those readings is always less than the reading of a clock measuring directly from A to C. In the ordinary space-triangle any two sides are together greater than the third side; in the time-triangle two sides are together less than the third side.
Rogers 1922	He showed that the "pure time-triangle" C, A, B (in their proper time order) satisfies the relation $\cosh C={\tfrac {\alpha ^{2}+\beta ^{2}-\gamma ^{2}}{2\alpha \beta }}$, where $\cosh C$ denotes the unit-scalar product of the vectors CA, CB, and $\alpha ,\beta ,\gamma$ the real and positive intervals BC, CA, AB. Since $\alpha >\beta$ and $\cosh C>1$, it follows that $\alpha >\beta +\gamma$. That is, "the greatest side of pure time-triangle is greater than the sum of the other two sides". It follows at once that the stationary value of the proper time integral is an "absolute maximum".

Negligibility of proper acceleration

Author	Early examples
Einstein 1905-1918	In 1905, Einstein used velocity time dilation $\tau =t{\sqrt {1-\left({\frac {v}{V}}\right)^{2}}}$ to derive the retardation of a clock performing a round-trip with constant speed $v$ along a polygonal path or a continuously curved line, without mentioning any influence of acceleration at turnaround. In 1911 (published 1912), Einstein said that special relativity doesn't say anything about what happened to the clock's pointer position during the acceleration that changes the clock's direction along the round-trip, yet the influence of this change must be getting smaller the longer the clock is moving uniformly, i.e. the longer one chooses the dimensions of the path. In an unpublished manuscript on special relativity from 1912, he pointed out that any influence of acceleration during the round-trip experiment, can be neglected if one makes the time of acceleration negligible with respect to the total time of motion along the polygonal path. In a letter from April 1914, Einstein showed that any finite acceleration at turnaround during the round-trip experiment can only influence the clock in a finite way, thus it can be neglected by minimizing the time of acceleration with respect to the time of uniform translation. So it must be concluded that the clock is retarded at reunion after traveling on a polygonal path. During a conversation in May 1914, Einstein is reported to have replied that the accelerations during the round-trip are "irrelevant for the amount of the time difference". (Compare with § Einstein 1914b-AC) In his famous "Dialog about Objections against the Theory of Relativity" from 1918, Einstein pointed out that any effect of velocity changes at turnaround must be limited, thus the traveling clock must be retarded at reunion due to time dilation if one makes the path AB and back along the round-trip long enough. (Compare with § Einstein 1918-AC)
Wiechert 1911	He demonstrated that differential aging along the round-trip cannot be caused during the passage from one velocity to another (i.e. acceleration) at turnaround, because the same result also follows when both A and B experience the same velocity changes with respect to another frame, only with the difference that B has relative velocities $+u$ and $-u$ for a long time, while A is brought after a short time from relative velocity $+u$ to relative rest at which it remains a long time, and then it is brought to relative velocity $-u$ for a short time. He was probably the first to use an example in which both accelerate with same magnitude.
Laue 1913	He showed that the problem of the influence of acceleration at turnaround in the round-trip experiment, can be eliminated by arbitrarily enlarging the time in inertial motion. This is the same argument as given in § Einstein 1911-VA. The Einstein-Laue argument was also used by others such as Thirring (1921) or Born (1921).
Lorentz 1913	He pointed out that any effect of acceleration on the traveling clock at turnaround, can be separated from the time dilation effect since only the latter depends on the distance traversed along the round-trip. Similarly, Pauli (1921) stated that the arising infinitesimal accelerations at turnaround are certainly independent of the total travel time and therefore easy to eliminate.

Relay (three brothers) experiment

Author	Early examples
Grünbaum 1911	He discussed a one-way time dilation experiment in which the first clock is set into motion from the origin and then moving to the second clock. He argued that one can avoid the problem of acceleration experienced by the first clock when set into motion, by replacing it with a third clock that is already in motion with constant velocity and is synchronized at the origin with the first clock. While Grünbaum didn't discuss round-trip experiments, his introduction of a third clock in order to avoid acceleration is the basis of the three-brother experiment.
Wiechert 1920-1922	In 1920 (published 1921), Wiechert explained how to completely remove acceleration from the round-trip experiment: Bodies A, B, C move undisturbed and non-accelerated in different directions. A and B pass each other at time (1), B and C pass each other at a later time (2), and C and A finally pass each other at an even later time (3). So in this setup, the condition of C is the continuation of the condition of B. On any of the three bodies one can count the oscillations of light of a certain spectral-line, in which case relativity predicts that the combined sum of all oscillations on B+C is smaller than the number of oscillations on A alone. 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+C only had time for one meal, there were arbitrarily many generations on A who follow after each other by death and birth. In 1921 (published 1922), Wiechert extended his previous acceleration-free round-trip experiment to an arbitrary number of non-accelerated bodies $B_{1}$, $B_{2}$, ..., which constitutes a "relay" (German: Stafette) starting from body A and back again. The first B passes A and moves away, and after some time the last B comes back to A. Since any B body continues the fate of the previous one, all bodies $B_{1}$, $B_{2}$, ..., combined have emitted fewer oscillations than A alone during the relay race. Wiechert pointed out that instead of light oscillations one can also choose the aging of life forms. Such relay experiments were later independently rediscovered in English language papers such as by Lange (1927) in which the brothers synchronize their times when they pass each other (“three brother experiment”).

Acceleration as asymmetry indicator

While it was known that any direct influence of w:proper acceleration on clocks can be neglected in the computation of the inertial frame of the stay-at-home twin (see previous section § Negligibility of proper acceleration), the very fact that only one of them is accelerating is still useful as an asymmetry argument in order to show that there is no contradiction to the relativity principle.

Author	Early examples
Langevin 1911	He derived differential aging in the round-trip experiment using the proper time integral along worldlines (see § Langevin 1911-PT) and used acceleration as an asymmetry indicator: The result of the round-trip experiment is "another example of the absolute character of acceleration" in which the "asymmetry occurred because only the traveler, in the middle of his journey, has undergone an acceleration that changes the direction of his velocity".
Sommerfeld 1913	After he showed (see § Sommerfeld 1913-PT) that retardation of time in the round-trip experiment derived from the proper time integral rests on the assumption that the clock's rate only depends on its momentary velocity (now called "clock hypothesis"), he used acceleration as an asymmetry indicator: There is no contradiction to the relativity principle since one of the clocks has to be accelerated in order to come back, thus the retardation in the round-trip experiment does not demonstrate "motion", but "accelerated motion".
Lorentz 1913	After he derived differential aging in the round-trip experiment from velocity time dilation and pointed out the negligibility of proper acceleration for the computation, he used acceleration as an asymmetry indicator: There is no contradiction to the relativity principle, since one of them changes velocity and accelerates; the relativity principle does not require symmetry between inertial and non-inertial observers.
Einstein 1914-1920	During a conversation in 1914, Einstein is reported to have said that moving clock B is retarded because it was accelerating in contrast to clock A; while those accelerations are irrelevant for the amount of the time difference, their presence nevertheless cause B to fall behind ("accelerated motions are absolute"). In his famous "Dialog about Objections against the Theory of Relativity" from 1918, Einstein pointed out the negligibility of velocity changes from the viewpoint of an inertial frame (see § Einstein 1918-VA). Then he used acceleration as an asymmetry indicator in order to show, that there is no contradiction to the relativity principle, because relativity only predicts the equivalence of non-accelerated inertial frames: "only K is such a frame while K' is temporarily accelerated, thus the retardation of U2 with respect to U1 cannot be used to construe a contradiction against the theory." Einstein is reported to have said in an interview from 1920: German original English translation Bei diesen Zwillingen, erklärte Einstein, haben wir zunächst eine Gefühls-Paradoxie vor uns. Eine Denk-Paradoxie würde indeß nur dann vorliegen, wenn sich für das Verhalten der beiden Geschöpfe kein zureichender Grund anführen ließe. Dieser Grund für das Jüngerbleiben des A ergibt sich vom Gesichtspunkt der speziellen Relativitätstheorie aus der Tatsache, daß das betreffende Geschöpf — und nur dieses — Beschleunigungen erlitten hat. In the case of these two twins," Einstein declared, "we have merely a paradox of feeling. It would be a paradox of thought only if no sufficient ground could be suggested for the behaviour of these two creatures . This ground, which counts for the comparative youth of A, is given, from the point of view of the special theory of relativity, by the fact that the creature in question, and only this creature, has been subject to accelerations." In a discussion from 1922, Einstein is reported to have said that there is no contradiction in the round-trip experiment (in terms of a train leaving the station and returning later): The relativity principle is not applicable to this case, because the train is not in a Galilean system (i.e. inertial frame) any longer during the period of velocity change at turnaround, i.e. the ensemble of two frames having velocities in opposite direction is not an inertial frame. There is no reciprocity between a frame that changes direction and one that doesn't.

Frame distribution as asymmetry indicator

Because any direct influence of proper acceleration on the traveling clock at turnaround can be neglected (see § Negligibility of proper acceleration), the importance of § Acceleration as asymmetry indicator is limited to the mere fact that it reveals that only the traveler was in a non-inertial frame as only he changed his inertial frames, thus instead of emphasizing the occurrence of proper acceleration at turnaround, it's possible to describe the asymmetry more geometrically by emphasizing the different distribution of inertial frames of the twins along their worldlines.

Author	Early examples
Laue 1911-1913	In 1911/12, he pointed out that during the time of separation, that clock is most advanced which was at rest in an inertial frame all the time; namely there is always one, and only one inertial frame, in which the locations of separation and re-encounter lie in the same geometric point. He clarified this fact by alluding to different paths in spacetime (compare with § Laue 1911/12-PT). In 1912/13, he argued that in the round-trip experiment, we indeed can decide, which one of the clocks was steadily at rest in one and the same reference system, and which one was in the meantime at rest in two or more such systems. Among them there is of course a real physical difference. He clarified this fact by alluding to different paths in spacetime (compare with § Laue 1912/13-PT). In 1913 Laue pointed out: German original English translation Aber nach unseren Voraussetzungen ruht während der Zeit der Trennung die erste Uhr in einem berechtigten Bezugssystem, die zweite hingegen ruht zwar sowohl bei der Hin- wie bei der Rückbewegung in berechtigten Bezugssystemen, aber notwendig in zwei verschiedenen. Deshalb unterscheiden sich beider Schicksale physikalisch. Ließe man die zweite Uhr in der ihr anfangs erteilten Bewegung und schickte man ihr dafür die erste Uhr nach einiger Zeit mit größerer Geschwindigkeit nach, so würde beim Zusammentreffen die erste gegen die zweite zurückgeblieben sein; denn jetzt hat die erste während der Trennung in zwei verschiedenen Systemen geruht. (Footnote: Dem naheliegenden Einwand, daß wir über den Gang einer Uhr während eines Geschwindigkeitswechsels nichts aussagen können, begegnet man am einfachsten mit dem Hinweis, daß man die Zeiten der gleichförmigen Bewegung beliebig groß gegen die der Beschleunigung machen kann.) However, by our presuppositions, one clock is at rest in one valid reference system during the time of separation, while the second one is at rest in valid reference systems both during the forward- and the backward motion, but necessarily in two different ones. Therefore the two fates differ physically. If we would let remain the second clock in the motion which was given to it at the start, and if we send after it the first clock after some time by a greater velocity, then at the encounter the first one would be retarded with respect to the second one; since now it was the first one that was at rest in two different systems during the separation. (Footnote: The objection which is near at hand, that we cannot say anything about the rate of a clock during a velocity change, can be met most simply by the allusion, that we can render the times of uniform motion arbitrarily great with respect to acceleration..)
Bloch September 1918	He represented the frames with three movable slots K, K' and K”, provided with hooks on which one can hang clocks at the origins of K and K'; while one clock always hangs on a hook of slot K, the other clock moved away with K' and after some time was transferred (neglecting any effect of acceleration) by a mechanical device to slot K” that moves in the other direction, by which it comes back; there is no contradiction to the relativity principle, as one clock rested in one inertial frame while the other one rested in two such frames.

Perspective of the traveler

Author	Early examples
Langevin 1911	After deriving differential aging from the proper time integral in § Langevin 1911-PT and using human beings in § Langevin 1911-HU, he described the perspectives of both observers using light signals and the Doppler effect. When they separate they see each other live 200 times slower, while at return they see each other live 200 times faster. So from the explorer's viewpoint, in the first year he sees the Earth perform the actions of two days, while in the second year he sees the Earth perform the actions of two centuries. The asymmetry can be seen by noticing, that the observer on Earth in 200 years sees the explorer performs the actions of 1 year. Then the explorer turns around, after which the observer on Earth in 2 days sees the projectile perform the actions of another year. Langevin used $v=c\left(1-{\tfrac {1}{20000}}\right)$, producing Lorentz factor $\gamma \approx 100$ and Doppler factor ${\sqrt {\tfrac {c+v}{c-v}}}\approx 200$.
Lorentz Lectures published in 1913 Similar treatment in 1914	Described the round-trip experiment in terms of inertial observer A (equipped with clock K) and traveling observer B (equipped with clock K'). In the frame of A, clock K' is retarded with respect to K at reunion due to time dilation. He then described the perspective of the traveling observer B by using two-way propagation of light from K' to K and back to K', leading to three periods defined by the moment of B's turnaround: In the first period the light signals return to K' before turnaround; in the second period the signals are emitted before turnaround and return after turnaround; in the third period emission and return of the signals are both happening after turnaround. Lorentz showed that K is time dilated by a factor of ${\sqrt {1-v^{2}/c^{2}}}$ with respect to K' in the first and third period, but in the second period K is ticking faster than K' by a factor of ${\sqrt {\tfrac {c+v}{c-v}}}$ which overcompensates the dilation in the other periods and explains, even from the perspective of B, why K' is retarded with respect to K at reunion. In a review of the German translation of Lorentz's book, Einstein (1914) didn't directly mention Lorentz's treatment of the twin paradox, but he wrote that nobody who is seriously interested in relativity should neglect to read that book. Pauli (1921) refers to Lorentz's book as one of three papers that analyze the twin paradox more closely.
Einstein 1916-1920	In a lecture from 1916, of which only an abstract was published, Einstein spoke about the "clock paradox of special relativity from the standpoint of w:general relativity." In a letter from September 1918, Einstein showed that general relativity makes the inertial frame K and and the accelerated frame K' of the clocks in the round-trip experiment "equally justified", explaining the time difference in K' by combining the influence of velocity and gravitational potential on clocks. In his famous "Dialog about Objections against the Theory of Relativity" from November 1918, aimed at clarifying misconceptions of the clock paradox, he explained that there is no paradox in special relativity because there is no symmetry between clock U1 at rest in inertial frame K and clock U2 at rest in accelerated frame K' (see § Einstein 1918-AC). Yet w:general relativity and the w:equivalence principle allow the treatment of this problem also from the standpoint of frame K', where clock U2 remains at rest all of the time while U1 makes the following movements: (1) It is accelerated by a homogeneous gravitational field in the negative direction, (2) it moves with constant velocity $-v$, (3) it is accelerated in the positive direction until it turns around and comes by with constant velocity $+v$, (4) it moves with velocity $+v$, (5) it is accelerated in the negative direction until it stops. Clock U1 is retarded with respect to U2 in periods 2) and 4) due to velocity time dilation, but this retardation is overcompensated by the faster rate of U1 during period 3), because U1 is at a higher gravitational potential. He argued that the computation (which he didn't provide) shows that the advance of U1 in period 3) is double its retardation during periods 2) and 4). Einstein concluded that by this consideration "the paradox is completely resolved". Using w:Mach's principle, he pointed out that the gravitational field in K' might be induced by the masses of the universe that are accelerated in this frame. In a letter to Einstein from December 1918, Jakob doubted the result that the advance in period 3) is double the retardation during periods 2) and 4). Einstein responded by letter, in which he used the gravitational time dilation factor $1+\Phi /c^{2}$ in K' in order to show that U1 at distance $l$ is advancing by $\Phi /c^{2}=2vl/c^{2}$ in period 3), which is indeed the double of approximated delay $vl/c^{2}$ caused by velocity time dilation during periods 2) and 4). Einstein is reported to have said in an interview from 1920, that while acceleration explains the age difference between the stationary twin B and the traveling twin A in terms of special relativity (see § Einstein 1920-AC), the "proper" description in terms of general relativity is as follows: German original English translation Eine tiefere Erfassung des Grundes ist indeß nur auf dem Boden der „Allgemeinen Relativitätstheorie" zu erlangen, die uns erkennen läßt, daß von A aus beurteilt ein Zentrifugalfeld existiert, von B aus betrachtet aber nicht; und dieses Feld hat einen Einfluß auf den relativen Ablauf und die Raschheit der Lebensvorgänge. A proper grasp of the reason is furnished only when we adopt the general theory of relativity, which tell us that, from the point of view of A, a centrifugal field exists, whereas it is absent from the point of view of B. This field exerts an influence on the relative rate of happening of the events of life." a) Einstein's explanation was quickly adopted in the textbooks of Bloch (1920), Pauli (1921), Kopff (1921), Bollert (1921), Born (1921), expressing the view that general relativity is "necessary" to provide the "complete" solution of the twin paradox. b) From a modern standpoint, however, Einstein's explanation has nothing to do with general relativity, but is rather an application of accelerated frames and "pseudo"-gravitational fields to flat Minkowski space of special relativity.
Thirring April 1921	He described the round-trip experiment by using two platforms K (clock A) and K' (clock B) each equipped with rows of clocks. He first demonstrated the symmetry of time dilation and the mutual relativity of simultaneity on the platforms and its effect on clock synchronization. The K clocks that B passes are all advanced because of $t'=\gamma \left(t-vx/c^{2}\right)$, and the same is true after turnaround since only the direction of velocity has to be changed in the Lorentz transformation $t'-t'_{0}=\gamma \left(t+vx/c^{2}\right)$ leading to the effect of clock desynchronization, where $t'_{0}$ is a constant depending on which clock one uses as standard for the new synchronization. He graphically showed using Minkowski diagrams, that this simultaneity jump due to desynchronization amounts to double the velocity time dilation during the inertial phases, explaining why A is more advanced than B at reunion. Using clock B as synchronization standard, Thirring's constant is given by $t'_{0}=2l\gamma v/c^{2}=2t\gamma v^{2}/c^{2}$ with $l=vt$ as position of turnaround. A similar explanation was subsequently given by Langevin (1922).

Curved spacetime

While the previous examples are defined in flat Minkowski spacetime and therefore can be fully discussed in terms of special relativity, general relativity is required when w:spacetime curvature in the presence of mass and energy cannot be neglected any more.

Author

Early examples

Becquerel 1922

After defining gravitational time dilation $d\tau ={\sqrt {1-{\tfrac {2GM}{c^{2}r}}}}dt$ in terms of the w:Schwarzschild metric around a material center, he discussed the following round-trip experiment: There are two identical clocks A and B placed next to each other, at a point very far from the material center, initially marking the same time $t$. Let us transport clock A to a point where the field is more intense, at a distance $r$ from the center; this clock will measure time $\int d\tau$ which is shorter than $\int dt$, thus it will run more slowly. If we bring clock A back to clock B, we will have to note that it is retarded with respect to B.

Paradoxical?

German original of Laue (1911/12):	English translation
Unter all den paradox erscheinenden Folgerungen aus der Zeittransformation der Relativitätstheorie gibt es wohl keine, gegen welche sich der natürliche Menschenverstand bei jedem, der der Sache noch ungewohnt ist, so sehr sträubt, wie gegen die, daß die Zeitangabe einer Uhr von ihrem Bewegungszustand abhängen soll. Schon in seiner grundlegenden Arbeit hat Einstein diese Paradoxie auf die Spitze getrieben in einem Gedankenexperiment, welches neuerdings von Langevin in einem auch sonst sehr lesenswerten Vortrage besonders hübsch erläutert worden ist.	Of all apparently paradox consequences that stem from the time-transformation of the theory of relativity, there is probably none against which the common sense of anyone who is still unfamiliar with the matter is more reluctant, than the one according to which the time indication of a clock shall be dependent on its state of motion. Already in his fundamental paper, Einstein has driven this paradox to the extreme by a thought experiment, recently explained in a very nice way by Langevin in a lecture that is also very readable in other respects.
Laue was probably the first to denote the round-trip experiment as paradoxical (even though he pointed out that there are no real contradictions). Subsequently, Gruner (1912) and others including Einstein (1918) explicitly used the expression "clock paradox" (French: Paradoxe des horloges, German: Uhrenparadoxon), whereas Seeliger (1913) spoke of the "familiar Einstein-Langevinian paradox" (German: "bekannte Einstein-Langevinsche Paradoxon").

Misunderstandings

German original by Berg (1910):	English translation
Im Punkte $x=0$ des Systems S befinde sich eine Uhr, eine andere im Punkte $x'=0$ von S'. Diese zweite bewege sich mit S' bis zum Punkte $x=a$, kehre dort um und bewege sich nun mit der Geschwindigkeit $v$ zurück bis zum Punkte $x=0$. Welche Zeit müssen beide Uhren in dem Moment angeben, wo sie sich wieder treffen? Wir beantworten diese Frage zunächst vom Standpunkt des Beobachters in S. Die Uhr in $x'=0$ hat sich mit der Geschwindigkeit $v$ bis zum Punkte $x=a$ bewegt; dazu brauchte sie die Zeit $\tau ={\tfrac {a}{v}}$. Zum Rückweg ist dieselbe Zeit nötig. Nach der Zeit $2\tau =2{\tfrac {a}{v}}$ ist die Uhr also wieder im Punkte $x=0$ angelangt. Wir stellen uns nun auf den Standpunkt des Beobachters in S'. Für diesen führt nach dem Relativitätsprinzip das System S genau dieselben Bewegungen aus wie das System S' für den Beobachter in S, nur in entgegengesetzter Richtung. Die Zeit bis zum Zusammentreffen beider Uhren ist also im System S' ebenfalls gegeben durch $2\tau =2{\tfrac {a}{v}}$. Betrachtungen, die auf anschauliche Vorstellungen, wie Nachgehen von Uhren, gestützt sind, führen hier leicht zu Irrtümern, von denen auch die Fachlitteratur nicht frei ist.	There is a clock at point $x=0$ of system S, and another one at point $x'=0$ of S'. The second one moves together with S' until point $x=a$, turns around and now moves back with speed $v$ to point $x=0$. Which time must both clocks indicate at the moment at which they encounter again? We answer this question at first from the standpoint of the observer in S. The clock at $x=0$ has been moving with speed $v$ until point $x=a$, for which it required time $\tau ={\tfrac {a}{v}}$. The same time is required for the way back. After time $2\tau =2{\tfrac {a}{v}}$ the clock has thus arrived again at point $x=0$. Let's now take the standpoint of the observer in S'. In his view in accordance with the relativity principle, system S is conducting exactly the same motions as those of system S' with respect to the observer in S, only in opposite direction. Thus the time until the meeting of both clocks is given by $2\tau =2{\tfrac {a}{v}}$ in system S' as well. Considerations based on illustrative notions, such as the retardation of clocks, easily lead to mistakes at this place, of which also the professional literature isn't free.
Berg was probably the first to turn the relativity principle against asymmetric aging in the round-trip experiment, claiming that both clocks must indicate the same time at reunion. See w:Twin paradox as well as sections §§ Acceleration as asymmetry indicator, Frame distribution as asymmetry indicator, and Perspective of the traveler for the solution of that problem.
German original by Wiechert (1911)	English translation
Even though he correctly derived differential clock aging in the round-trip experiment, he claimed that effects like time dilation are "apparent" if one admits Einstein's "unconditional" relativity principle in which there is no aether and all "strides" (i.e. non-accelerated motions) are physically equivalent, but they are "real" if one admits the existence of an aether in the framework of a "conditional" relativity principle in which all strides are physically non-equivalent or anisotropic. This led him to the following interpretation of the clock paradox:
[...] so muß am Schluß des Versuches B in seinem Fortschritt gegenüber A im Verhältnis $1:{\sqrt {1-u^{2}/c^{2}}}$ zurückgeblieben sein. Und dieses Zurückbleiben ist unbedingt reell, denn die beiden Gebilde A und B können ja unter gleichen Umständen unmittelbar beieinander verglichen werden. Hier ist es ganz sicher ausgeschlossen, an einen Schein zu glauben, der durch unsere Auffassung der Zeit bewirkt wird. So ist denn also auch die Folgerung unabwendbar, daß für den Verlauf der Weltvorgänge die Schreitungen nicht gleichwertig sind, und damit sind wir von neuem zu einem Schluß gekommen, welcher der Unbedingtheit des Relativitätsprinzipes durchaus widerspricht. [...] Man kann den Versuch noch mannigfach variieren, z. B. so, daß A ebenso wie B zwei verschiedene Schreitungen, $+u$ und $-u$, nacheinander inne hat. Wird dann zu A der Wert $u_{1}$, zu B der Wert $u_{2}$, zugeordnet, so muß der Vergleich von A und B am Schluß des Versuches ergeben, daß B oder A in seinem Fortschritt zurückgeblieben erscheint, je nachdem die Schreitungen $+u_{1}$, $-u_{1}$, oder $+u_{2}$, $-u_{2}$ weiter auseinanderliegen. Vielleicht ist gerade diese Formulierung des Satzes besonders geeignet, um die Ungleichwertigkeit der verschiedenen Schreitungen klar und deutlich zu zeigen.	[...] thus B's progress must be retarded with respect to A's in the ratio $1:{\sqrt {1-u^{2}/c^{2}}}$ at the end of the experiment. And this retardation is definitely real, since both bodies A and B indeed can be immediately compared side by side under the same conditions. Here it is certainly excluded to believe that this is an appearance due to our conception of time. Thus the consequence is unavoidable too, that the strides are not equivalent in the course of the world processes, and therefore we again came to a conclusion that completely contradicts the unconditionality of the relativity principle. [...] One can vary this experiment in many ways, for instance, so that A in the same way as B successively undergoes two different strides $+u$ and $-u$. If we apply the value $u_{1}$ to A and $u_{2}$ to B, then the comparison of A and B at the end of the experiment must give the result, that B or A is retarded in its progress depending on whether the strides $+u_{2},-u_{2}$ or $+u_{1},-u_{1}$ are further apart. Probably it is precisely this formulation of the theorem that is particularly suitable to demonstrate the non-equivalence of the different strides clearly and explicitly.
This interpretation was directly rebutted by Laue (1911/12) who demonstrated the geometrical meaning of differential aging in Minkowski space, see sections §§ Laue 1911/12-PT​ and Laue 1911/12-VA, showing that there is no need to assume non-equivalance or anisotropy of motions. Laue added, that as long as there is no experimental contradiction to the relativity principle, the question after the aether can be banned from physics and left to philosophy.
German original by Campbell (November 1911, published 1912)	English translation
After describing the round-trip experiment (as given by Wiechert) according to which the traveling clock B is retarded when it returns with respect to stationary clock A, he abandoned differential clock aging as follows:
Dieser Schluß ist nicht richtig. Die Beziehung zwischen $t$, der Ablesung an der Uhr auf A seitens des Beobachters auf A und $t'$, der Ablesung an der Uhr auf B seitens des Beobachters auf A, ist (unter der Annahme, daß zu Beginn des Versuchs $t=t'$ ist) $t'={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}\left(t-vz/c^{2}\right)$. Der Unterschied zwischen $t'$ and $t$ ist eine Funktion von $z$ und $v$ allein. Wenn man diesen Größen ihre früheren Werte wiedergibt, indem man die beiden Uhren wieder zur Koinzidenz bringt, während sie relativ zueinander ruhen, so geht der Unterschied zwischen $t'$ and $t$ wieder auf null zurück, gleichviel, welche Werte $z$ und $v$ während der Zwischenzeit gehabt haben mögen. Wenn an irgendeinem Punkte der Bahn die Geschwindigkeit von B relativ zu A eine endliche plötzliche Änderung erfährt, so erfährt auch der Wert von $v$ eine endliche plötzliche Änderung.	This conclusion is not correct. The relationship between $t$ as the reading on the clock on A by the observer on A, and $t'$ as the reading on the clock on B by the observer on A, is given by (assuming that $t=t'$ at the beginning of the experiment) $t'={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}\left(t-vz/c^{2}\right)$. The difference between $t'$ and $t$ is a function of $z$ and $v$ alone. If these quantities are given their previous values by bringing the two clocks back to coincidence during which they are at rest relative to one another, the difference between $t'$ and $t$ goes back to zero, no matter what values $z$ and $v$ may have had in the meantime. If at any point on the path the speed of B experiences a finite sudden change relative to A, then the value of $t'$ also undergoes a finite sudden change.
So Campbell claims that any time difference during the outbound path is wiped out during the inbound path. His mistake is obvious: Campbell is confusing coordinate differences stemming from the Lorentz transformation of events (which indeed depend on position and direction) with differences in clock aging derived from the proper time integral (which is accumulative and independent of position and direction.)
French original by Gruner (March 1912):	English translation
[...] deux personnes du même âge, se séparant dans des systèmes de « marche » très différents et retournant après un laps de temps assez long, constateront une différence d'âge très sensible. [...] le principe de relativité exige toujours la réciprocité parfaite des phénomènes entre deux systèmes qui possèdent un mouvement relatif. Si, dans l'exemple cité, les deux personnes du même âge se séparent avec une vitesse relative pour se retrouver plus tard, la constatation d'une différence d'âge sera parfaitement mutuelle : A dira positivement que B est resté en arrière dans son développement, et B affirmera avec le même droit que c'est A qui ne s'est pas développé assez vite. Ainsi le principe absolu de la relativité montre ses conséquences les plus extrèmes et il est clair que l'introduction de l’éther n'est plus en état de résoudre cette contradiction irréductible et inconcevable.	[...] two people of same age, separating into very different systems of motion and returning after a quite long period of time, will notice a very significant age difference. [...] the principle of relativity always requires the perfect reciprocity of the phenomenons between two systems that possess relative motion. When, in the cited example, the two persons of same age are separated by some relative velocity only to meet again later, the finding of an age difference will be perfectly mutual: A will positively say that B stayed behind in its development, and B will assert with same right that it was A who has not developed fast enough. By that, the absolute relativity principle shows its most extreme consequences and it is clear, that the introduction of the aether is no longer able to resolve this irreducible and inconceivable contradiction.
Gruner was probably the first to claim that combining the round-trip experiment with the symmetry of time dilation leads to the contradictory situation, that both must attribute younger age to one another at reunion. At the end of his paper, we also find the expression "clock paradox" (French: paradoxe des horloges). See w:Twin paradox as well as sections §§ Acceleration as asymmetry indicator, Frame distribution as asymmetry indicator, and Perspective of the traveler for the solution of that problem.