Triplet paradox
An extension of the twin paradox, the triplet paradox is another thought experiment in special relativity, this time involving identical triplets. One of these triplets, A, stays on Earth while the other two, B and C, make a journey in a high-speed rocket to some far off point in space in opposite directions before coming back to Earth. Upon their return, B and C find that they are the same age while A has aged more than both of them. However, triplets B and C would each actually measure the other and triplet A as being younger. This is due to each sibling's reference frame. Relative to each triplet, the other two are moving away from them at high speeds. It must be noted that, while A is the only triplet on Earth, each triplet is at rest in their own reference frame compared to their other two siblings. As is the case with the aforementioned Twin Paradox, the Triplet Paradox is the a result of an incorrect[1] application of our understanding of time dilation. Since Triplets B and C must speed up, slow down, and change direction over the course of their journey, they no longer travel in what is known as an inertial reference frame. Instead, both spacefaring triplets have to accelerate over the course of their respective journeys. Essentially, when the trips taken by B and C are treated as long, continuous trips as opposed to accounting for their acceleration, they end up with contradictory calculations when compared to the math conducted by A.
History[edit]
Much of what is known about special relativity, specifically time dilation in the case of the Twin and Triplet Paradoxes, stems from Albert Einstein's initial theories on the matter in his 1905. In this paper, Einstein initially suggested that if two clocks were synchronized and one was moved away at a high speed while the other stayed put, upon its return the moving clock would be lagging behind the one that stayed still.[2] Einstein (in 1905), Paul Langevin (in 1911),[3] and Max von Laue (in 1913) presented their own interpretations of time dilation, giving way to what is known as the Twin Paradox.
The first mention of the Triplet Paradox comes from Vladimir Alexandr Leus in a 2015 paper entitled "Triplet Paradox in Special Relativity and Discrepancy with Electromagnetism." Leus suggests an example where two siblings are moving at high speeds in opposite directions "in a manner symmetrical to the basic frame of reference staying at rest."[4] In the case described above, that reference frame is Earth. In his paper, Leus considers the Triplet Paradox a violation of the relativity of simultaneity, the concept that observers in inertial reference frames will disagree upon the times at which two events supposedly occurred.[5] This can lead to illegal implementation of Lorentz transformations, the method by which relative quantities can be measured from other reference frames.
Specific example[edit]
Assume A, B, and C are identical triplets. Triplets B and C board identical high-speed rockets and travel in opposite directions from Earth to a star system a distance d = 4 lightyears away at a speed v = 0.8c (or 80% the speed of light). For the sake of simpler calculation, assume their speeding up and slowing down time at the start and end of their trips is negligible. Triplet A stays on Earth during the entirety of their journey.
Below, their respective journeys will be analyzed, showing how the math will look when conducted properly:
Earth perspective (Triplet A)[edit]
Triplet A sees themself as staying still relative to their two siblings moving at the same speed for the same amount of time away from the Earth and back. Due to B and C traveling away from Earth and then returning, they must cover a distance of 2d on their journey. Thus, the total time A waits for their siblings on Earth is t = 2d/v = 10 years. Thus, triplet A will be 10 years older when their siblings return to Earth (or, when Earth returns to their siblings).[6] Next, solve for the Lorentz Factor = 1.66666667. The reciprocal of the Lorentz Factor, α, is the factor by which time will be dilated for B and C on their spaceship. α therefore is 0.6 meaning the siblings in space will only age 10*0.6 = 6 years.
Traveler perspective (Triplets B and C)[edit]
Relative to the passengers on board their respective spaceships, both Earth and the other spaceship will be moving away from (and later towards) them while they remain at rest. In triplet B's reference frame, A is moving at a speed of v = 0.8c. Thus, α will be the same for B as for A. The distance, d, between Earth and the star system B travels to is 4 lightyears. However, when moving at high speeds, this length contracts by a factor of α. This is not to say that the distance literally gets smaller, but rather the distance appears smaller by an observer at rest. The new distance, d' = α*d = 2.4 lightyears. The total time this trip should take each way is d/v = 2.4/0.8 = 3 years. The full trip would take 6 years, as was measured by A. When this calculation is not split into two separate trips, the siblings will have each measured different age values, creating the paradox.
References[edit]
- ↑ Serway, Raymond A.; Moses, Clement J.; Moyer, Curt A. (2004-04-15). Modern Physics. Cengage Learning. ISBN 978-1-111-79437-8. Search this book on
- ↑ Einstein, A. (1905). "Zur Elektrodynamik bewegter Körper". Annalen der Physik (in Deutsch). 322 (10): 891–921. doi:10.1002/andp.19053221004.
- ↑ Langevin, Paul (1911). "The Evolution of Space and Time". Scientia. 10 – via AMS Historica.
- ↑ Leus, Vladimir Alexandr (2015). "Triplet Paradox in Special Relativity and Discrepancy with Electromagnetism". American Journal of Modern Physics. 4 (2–1): 26–33. doi:10.11648/j.ajmp.s.2015040201.15.
- ↑ "Special Relativity Simultaneity". sites.pitt.edu. Retrieved 2022-11-04.
- ↑ Krane, Kenneth (2012). Modern Physics (3rd ed.). John Wiley & Sons Inc. pp. 44–47. ISBN 978-1-118-06114-5. Search this book on
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