In physics, the twin paradox is a thought experiment in special relativity involving twins, one of whom takes a space voyage at relativistic speeds and returns home to find that the twin who remained on Earth has aged more. This result appears puzzling because each twin sees the other twin as moving, and so, as a consequence of an incorrect Extract of page 23 Extract of page 21 and naive Extract of page 541 Extract of page 176 application of time dilation and the principle of relativity, each should paradoxically find the other to have aged less. However, this scenario can be resolved within the standard framework of special relativity: the travelling twin's trajectory involves two different inertial frames, one for the outbound journey and one for the inbound journey. Extract of page 203 Another way to understand the paradox is to realise the travelling twin is undergoing acceleration, thus becoming a non-inertial observer. In both views there is no symmetry between the spacetime paths of the twins. Therefore, the twin paradox is not actually a paradox in the sense of a logical contradiction.
Starting with Paul Langevin in 1911, there have been various explanations of this paradox. These explanations "can be grouped into those that focus on the effect of different standards of simultaneity in different frames, and those that designate the acceleration [experienced by the travelling twin] as the main reason". Max von Laue argued in 1913 that since the travelling twin must be in two separate inertial frames, one on the way out and another on the way back, this frame switch is the reason for the ageing difference. Explanations put forth by Albert Einstein and Max Born invoked gravitational time dilation to explain the ageing as a direct effect of acceleration. However, it has been proven that neither general relativity,Extract of page 207David Halliday et al., The Fundamentals of Physics, John Wiley and Sons, 1997Paul Davies About Time, Touchstone 1995, ppf 59. nor even acceleration, is necessary to explain the effect, as the effect still applies if two astronauts pass each other at the turnaround point and synchronise their clocks at that point. The situation at the turnaround point can be thought of as where a pair of observers, one travelling away from the starting point and another travelling toward it, pass by each other, and where the clock reading of the first observer is transferred to that of the second one, both maintaining constant speed, with both trip times being added at the end of their journey.
History
In his famous paper on special relativity in 1905, Albert Einstein deduced that for two stationary and synchronous clocks that are placed at points A and B, if the clock at A is moved along the line AB and stops at B, the clock that moved from A would lag behind the clock at B. He stated that this result would also apply if the path from A to B was polygonal or circular. Einstein considered this to be a natural consequence of special relativity, not a paradox as some suggested, and in 1911, he restated and elaborated on this result as follows (with physicist Robert Resnick's comments following Einstein's):. via August Kopff, Hyman Levy (translator), The Mathematical Theory of Relativity (London: Methuen & Co., Ltd., 1923), p. 52, as quoted by G.J. Whitrow, The Natural Philosophy of Time (New York: Harper Torchbooks, 1961), p. 215.
In 1911, Paul Langevin derived differential ageing in round-trip experiments using portions of matter such as radium, showing by integration of the proper time of a co-moving accelerated observer, that the time experienced by that observer is shorter than for an observer in uniform motion, thus radium in motion is decaying at a slower rate. He went on to give a "striking example" by describing the story of a traveller making a trip at a Lorentz factor of (99.995% the speed of light). The traveller remains in a projectile for one year of their time, and then reverses direction. Upon return, the traveller will find that, having aged two years, 200 years have passed on Earth. During the trip, both the traveller and Earth keep sending signals to each other at a constant rate, which places Langevin's story among the Doppler shift versions of the twin paradox. The relativistic effects upon the signal rates are used to account for the different ageing rates. The asymmetry that occurred because only the traveller underwent acceleration is used to explain why there is any difference at all, Extract of page 39 Extract of page 39 because "any change of velocity, or any acceleration has an absolute meaning".; English translation The Evolution of Space and Time on Wikisource Langevin's acceleration argument was subsequently also used by others such as Arnold Sommerfeld (1913),See p. 71 where he said that the result "does not indicate motion, but rather accelerated motion" in: Einstein (1918) or Max Born (1921) in order specifically to point out that there is an asymmetry between the clocks.
Neither Einstein nor Langevin considered such results to be problematic: Einstein (1905) only called it "peculiar" while Langevin presented it as a consequence of absolute acceleration."We are going to see this absolute character of the acceleration manifest itself in another form." ("Nous allons voir se manifester sous une autre forme ce caractère absolu de l'accélération."), page 82 of Langevin1911 They argued that, from the time differential illustrated by the story of the twins, no self-contradiction could be constructed. In other words, none of them saw the story of the twins as constituting a challenge to the self-consistency of relativistic physics. Max von Laue (1911) was apparently the first. to use the expression "paradox" in relation to this experiment by writing:; See also English translation Two Objections Against the Theory of Relativity and their Refutation on Wikisource
Laue (1911, 1913) went on to elaborate on Langevin's explanation:; See also English translation The Principle of Relativity, Second edition, Part III on Wikisource Using Hermann Minkowski's spacetime formalism, he demonstrated that the world lines of the inertially moving bodies maximise the proper time elapsed between two events, which was also pointed out in the textbooks of Hermann Weyl (1918) or Wolfgang Pauli (1921). Regarding the , Laue wrote in 1913 that the asymmetric ageing is completely accounted for by the fact that the astronaut twin travels in two separate frames, while the Earth twin remains in one frame, and the time of acceleration can be made arbitrarily small compared with the time of inertial motion.; See also English translation The Principle of Relativity (article) on Wikisource Even before Laue, Emil Wiechert (1911) pointed out that there can be a time difference at reunion even when both clocks experience the same velocity changes or accelerations,See p. 745f. general derivation; p. 757f. space travel: and in 1921 Wiechert removed any acceleration by introducing the "relay" or "three-brother" approach, in which the travelling twin transfers their clock reading to a third one, travelling in the opposite direction.See figures 1a and 1b on p. 25ff for the introduction of the "relay" (German: Stafette) using several non-accelerated clocks:
In addition to Langevin's Doppler analysis, several methods have been invented to specifically show in which way the symmetry of time dilation is broken during turnaround from the perspective of the traveler: Hendrik Antoon Lorentz (1913) described that traveler's perspective in terms of two-way signals (radar) using three periods: In the first and the last, the stay-at-home clock moves at constant velocity and is therefore time dilated, yet in the middle period it apparently ticks faster which overcompensates the other periods and explains, why the traveler's clock is retarded at reunion with respect to the stay-at-home clock even from the traveler's own perspective.See pp. 37f, 55ff in: ; German translation on pp. 31f, 47f in: See early solutions in: ; A shorter version appeared in . Alternatively, Hans Thirring (1921) used the Lorentz transformation to show that relativity of simultaneity leads to desynchronization of all the traveler's clocks during turnaround; this leads to an apparent forward jump of the stay-at-home clock which overcompensates its time dilation during constant velocity motion, all of which was visualized by Thirring using Minkowski diagrams.See p. 209ff in: The same result was already before obtained by Einstein (1918), who used the equivalence principle to show that a homogeneous gravitational field appears in the rest frame of the traveler at turnaround, which is accompanied by gravitational time dilation; since the "free-falling" stay-at-home clock is at a location of higher potential during turnaround, it advances at a faster rate which overcompensates its retardation during constant velocity motion.Einstein, A. (1918), "Dialog about objections against the theory of relativity", Die Naturwissenschaften 48, pp. 697–702, 29 November 1918 Einstein's explanation was adopted in 1921 in the textbooks of Wolfgang Pauli and Max Born.See p. 558f (general description); p. 624f (proper time); p. 713f (accelerated frames); in: ; English translation by G. Field (1958): Theory of RelativitySee pp. 190f. (special relativity), 250f (EP) in: ; See also the English translation of the third edition by H. Brose (1924): Einstein's theory of relativity, pp. 214ff, 282ff.
Eventually, Hermann Weyl (1918) replaced the clocks and travelers explicitly by "twins", writing:See p. 147f. in ; English translation of the 4th edition by H. Brose (1921): Space—Time—Matter, pp. 278f.
Specific example Consider a space ship travelling from Earth to the nearest star system: a distance years away, at a speed (i.e., 80% of the speed of light).
To make the numbers easy, the ship is assumed to attain full speed in a negligible time upon departure (even though it would actually take about 9 months accelerating at 1 g to get up to speed). Similarly, at the end of the outgoing trip, the change in direction needed to start the return trip is assumed to occur in a negligible time. This can also be modelled by assuming that the ship is already in motion at the beginning of the experiment and that the return event is modelled by a Dirac delta distribution acceleration.
The parties will observe the situation as follows: Extract of page 74 Extract of page 27
Earth perspective The Earth-based mission control reasons about the journey this way: the round trip will take in Earth time (i.e. everybody who stays on Earth will be 10 years older when the ship returns). The amount of time as measured on the ship's clocks and the ageing of the travellers during their trip will be reduced by the factor , the reciprocal of the Lorentz factor (time dilation). In this case and the travellers will have aged only when they return.
Travellers' perspective The ship's crew members also calculate the particulars of their trip from their perspective. They know that the distant star system and the Earth are moving relative to the ship at speed v during the trip. In their rest frame the distance between the Earth and the star system is years (length contraction), for both the outward and return journeys. Each half of the journey takes , and the round trip takes twice as long (6 years). Their calculations show that they will arrive home having aged 6 years. The travellers' final calculation about their ageing is in complete agreement with the calculations of those on Earth, though they experience the trip quite differently from those who stay at home.
Conclusion + Readings on Earth's and spaceship's clocks Event Earth(years) Spaceship(years) Departure 0 0 End of outgoing trip =Beginning of ingoing trip 5 3 Arrival 10 6