In special relativity, the history of the twin paradox chronicles the development of the famous thought experiment known as the twin paradox, which started in 1905 when Albert Einstein showed that a clock returning from a round-trip shows less elapsed time than a clock that remained at its place. By 1911, Einstein, Paul Langevin and Emil Wiechert had extended this result to life forms and human beings, with Hermann Weyl introducing twin brothers in 1918.
In 1911, Langevin, Wiechert and Max von Laue showed that asymmetric aging of clocks or life forms along closed paths directly corresponds to the fact, that the proper time is maximal along the straight worldlines of non-accelerated clocks. Laue pointed out that the relativity of time and asymmetric aging appears "paradoxical" to those who are still unfamiliar with relativity, while some even claimed that asymmetric aging contradicts the relativity principle, which was refuted by Laue himself and others such as Einstein (1918) who pointed out the meaning of the relativity principle and demonstrated the asymmetries between the twins.
Round-trip experiment (1905)
In 1905, Einstein derived the effect of time dilation which he applied to round-trips as follows:
From this there ensues the following peculiar consequence. [...] If one of two synchronous clocks at A is moved in a closed curve with constant velocity until it returns to A, the journey lasting seconds, then by the clock which has remained at rest the traveled clock on its arrival at A will be second slow. [Translation by Perret and Jefferies]
In a lecture from January 1911 (published November), Einstein extended this result to life forms:
If we placed a living organism in a box [...] one could arrange that the organism, after any arbitrary lengthy flight, could be returned to its original spot in a scarcely altered condition, while corresponding organisms which had remained in their original positions had already long since given way to new generations. For the moving organism, the lengthy time of the journey was a mere instant, provided the motion took place with approximately the speed of light. [Translation by Miller]
Two participants of this lecture, Rudolf Lämmel in April 1911 and Fritz Müller in October 1911, reported in newspaper articles that Einstein also talked about human observers making a journey near light speed who practically didn't age during the journey, yet as they return they meet later generations (Lämmel) or old men with gray beards (Müller). This experiment became particularly known through a lecture by Langevin in April 1911 (published July), who independently stated that a portion of matter conducting a closed cycle will have aged less between its departure and its return than if it had not been accelerating, thus radium will have decayed less during the round-trip than a stationary sample, and a human traveler flying in a rocket and returning after 200 years Earth time will experience only 2 years during the flight. Independently, Wiechert in lectures between March and May 1911 (published July/September) discussed several round-trip clock experiments, including an experiment in which a human observer traveled for 300 Earth years on a circular path away and back to Earth, while he himself only experienced 75 years. Similarities between such experiments and fictional stories such as Rip Van Winkle, Urashima Tarō or When The Sleeper Wakes were noticed by several authors.
The round-trip experiment was explicitly formulated in terms of twins, i.e. two life forms of equal age, by the following authors: Wiechert (1911) spoke about "two life forms that begin their life at the same time", Paul Gruner (1912) about "two persons of same age", Laue (1913) about "former agemates", Weyl (1918) about "twin brothers", Werner Bloch (1918) about "two humans of exactly same age", and Einstein (1920) about "twins". For instance, the description of Weyl reads:
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. [Translation by Henry Brose]
Maximal proper time (1911)
After Einstein (1905) derived asymmetric clock aging along a closed polygonal path directly from the time dilation formula, more general treatments based on the proper time integral introduced by Hermann Minkowski (1907) followed. Langevin (1911–1919) showed that the integration of proper time leads to the result, that the time experienced by an accelerated observer traversing a closed path is shorter than for an observer in uniform motion. Also Wiechert (1911) noticed that the time difference in round-trip experiments can "easily" be derived by applying the proper time integral. Laue (December 1911) showed that asymmetric clock aging directly corresponds to the geometrical property of spacetime that among all time-like intervals connecting two events, the straight worldline has the maximal proper time as indicated by the clock that remained at rest in a single inertial frame, so differential aging is only about the different ways by which two points in spacetime are connected. Consequently, Laue (1912/13) employed the inequality
,
in which the right-hand side represents the "maximal proper time" along a straight worldline, while the left-hand side represents smaller proper times of clocks along a broken worldline between the same events. This explanation was adopted by others such as Arnold Sommerfeld (1913), Weyl (1918), August Kopff (1921), Jean Becquerel (1922), with Wolfgang Pauli (1921) calling this the "four-dimensional formulation of the clock paradox".
In the simplest case where the traveler only once changes his direction in a very small time, Laue's inequality reduces to the inverse triangle inequality in Minkowski space formulated by Alfred Robb (1914), stating that the length of a particular side is longer than the combined lengths of the other sides, if all sides represent time-like intervals. In 1920, Robb gave a numerical example in which the time-like intervals AB=10, AC=3, CB=3 form the sides of triangle with the inequality (Fig. 1). Arthur Eddington (1922) described those relations as "time-triangles" formed by moving clocks.
Several authors employed Minkowski diagrams to graphically demonstrate the round-trip experiment: Alexander von Brill (1920) demonstrated that the broken worldline + of the traveling clock has a smaller proper time than the straight one of the stationary clock; even though the worldlines drawn on paper apparently indicate the inequality , the computation of proper time actually gives because the units on the traveler's axis are warped by . Pauli (1921) described the round-trip experiment in terms of two diagrams: In the first, two points P and Q are connected by a straight worldline and a broken worldline ; in the second, two points A and B are connected by a straight worldline and by a continuously curved one. The diagram of Max Born (1921) showed that at the outbound journey, traveler B experiences proper time interval OU on the inclined -axis while stationary observer A experiences proper time interval OQ on the vertical t-axis, with points U and Q located on the same hyperbola; at the inbound journey, however, observer B returns when his proper time is exactly the double of OU, whereas observer A returns when his proper time is more than the double of OQ.
Negligibility of proper acceleration (1913)
It was pointed out that the acceleration experienced by the traveler (i.e. the proper acceleration) is neglectable when computing the round-trip experiment from the viewpoint of the inertial frame of the stay-at-home twin: Sommerfeld (1913) showed that the application of the proper time integral to the round-trip experiment rests on the "unprovable assumption" (which he attributed to Einstein) that the time indicated by the traveling clocks only depends on its momentary velocity (clock hypothesis). Laue (1912/13), Weyl (1918), Pauli (1921) avoided all practical problems related to the application of that integral to the travelling clock by demanding that the strength of acceleration must be sufficiently small. Einstein (1911, 1914, 1918), Laue (1913), Hans Thirring (1921), Born (1921) showed that even if there were an unknown influence of proper acceleration on the traveling clock's readings, this wouldn't necessarily change the conclusion that the traveling clock is retarded with respect to the stay-at-home clock when they reunite, because any such influence must be finite and therefore can be made arbitrarily small and negligible with respect to the effect of time dilation by arbitrarily elongating the duration of constant velocity motion. The same fact was demonstrated in an alternative way by Hendrik Lorentz (1913) who pointed out that any effect of acceleration of 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, and also Pauli (1921) argued that any influence of acceleration at turnaround must be independent of the total travel time along the round-trip and is therefore easy to eliminate.
Wiechert (1911) showed that even when both clocks undergo the same velocity changes or accelerations in the round-trip experiment, asymmetric clock aging can arise when the clocks reunite: Clock A travels with for a short time, then comes to rest at which it remains for a long time, and then returns with , while clock B traveled with for a long time (see Fig. 2), in which case B will be retarded with respect to A at reunion. Likewise, Wiechert 1921 (published 1922) described the round-trip of clock B along a polygonal path away from stationary clock A and back again; then the experiment is repeated with exact same velocity changes as before, yet the size of the polygon should be arbitrarily increased; this implies that any influence on clocks during the velocity changes will affect the endresults of the two experiments in different proportion, which allows us to determine and eliminate that influence.
While discussing a one-way time dilation experiment in which the first clock had to be accelerated at the start in order to reach the second clock, (1911) showed that problems related to unknown influences of acceleration can be avoided by replacing the accelerated clock with a third, non-accelerated clock that is synchronized with the accelerated clock when they are momentarily co-located. Wiechert in 1920 (published 1921) applied the same idea to round-trip experiments by replacing the clocks or twins with three non-accelerated bodies A, B, C, on which the time is measured either by counting light oscillations or by determining the aging of life forms; A is passed by B that moves away in some direction, later B is passed by C that moves with same speed in the opposite direction, and finally C passes A; it turns out that the combined time measured on bodies B+C during those periods is smaller than the time measured on A alone. In 1921 (published 1922) Wiechert extended what he described as "relay" experiment (German: Stafette) to arbitrary many non-accelerated bodies , in which the first B passes A at the beginning of the experiment and the last B passes A at the end of the experiment; it turns out that the combined time measured on all B bodies is smaller than the time measured on A alone (see Fig. 3 for the special case of three B clocks). Variants of this thought experiment became known as the three-brother experiments in which the traveling clocks synchronize their times when they pass each other, as discussed by Luise Lange (1927) or Lord Halsbury (1957).