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Criticism to the Twin's Paradox

Criticism to the Twin's Paradox

Universal Journal of Physics and Application 15(1): 1-7, 2021 http://www.hrpub.org DOI: 10.13189/ujpa.2021.150101

Criticism to the Twin's Paradox

Luís Dias Ferreira

Colégio Valsassina, Lisbon, Portugal

Received January 18, 2021; Revised February 22, 2021; Accepted March 23, 2021

Cite This Paper in the following Citation Styles (a): [1] Luís Dias Ferreira , "Criticism to the Twin's Paradox," Universal Journal of Physics and Application, Vol. 15, No. 1, pp. 1 - 7, 2021. DOI: 10.13189/ujpa.2021.150101. (b): Luís Dias Ferreira (2021). Criticism to the Twin's Paradox. Universal Journal of Physics and Application, 15(1), 1 - 7. DOI: 10.13189/ujpa.2021.150101. Copyright©2021 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 International License

Abstract The so-called “twin’s paradox” is considered 1. Introduction an important issue in theory because it implies a profound understanding of structure. To this day, I have not found in the literature or And yet, since its original formulation in 1911 by Paul anywhere else a real satisfactory answer to the so-called Langevin, numerous alleged explanations for this “twin’s paradox”. I think it is still worthy to meditate on disturbing paradox have been produced; as it seems, this uncomfortable paradox because its solution would unsuccessfully. This remains a subject for heated debate. bring a more profound comprehension of Why? Because in all those explanations one tries to behavior and of how one measures coordinates in it; this is, reconcile the irreconcilable, this is, what seems to be a of Relativity itself. logical conclusion (based on the phenomenon of time The basis for the paradox lies in a “peculiar dilation) with what is simply unacceptable: how can it be a consequence” of Special Relativity (SR), stated more than difference in aging from twins without breaking the one century ago by himself in his fundamental equivalency between frames of coordinates? founding paper on the subject [1]: The purpose of this research is, first, to point out the basic “Consider in the points A and B of a coordinate flaws in the premises of the usual “explanations” and then frame K two at rest, supposing they work in to provide a consistent answer to the problem. It is proven synchrony to whomever observe them in the frame at here that there is no twin’s paradox and this despite the rest. Let us imagine now that we communicate to the reality of . Proceeding without prejudice, in A a movement with velocity along the simply following appropriate premises and mathematical straight line that lies both points, in the sense of B. equations, one finally discovers an astoundingly, When this clock arrives at B synchrony𝑣𝑣 no longer wonderfully coherent resolution to the problem, and this in exists: compared to the one which remained in B, the the frame of special relativity itself. The key to understand clock that has been moved presents a delay of and finally resolve this puzzling issue is relativistic 1/2 / (up to quantities of the fourth and asynchrony, particularly past and future permutation. higher 2orders),2 if we represent by the duration of Finally, the implications of this understanding, as can be the displacement.𝑡𝑡𝑣𝑣 𝑐𝑐 We see at once that this result still easily induced, go far beyond special relativity. If there is holds if the clock moves from A 𝑡𝑡to B along any no different aging in inertial frames, regardless of their arbitrary polygonal line, and this even when the relative velocity, should this conclusion also apply to points A and B coincide.” accelerated ones, this is, to ? Einstein admits that “the deduced result for a Keywords Special Relativity, Twin’s Paradox, Time polygonal line holds also for a continuously curved line”, Dilation, Relativistic Asynchrony even in the case of closed curves. All these conclusions are then expected to be verified in the modern particle accelerators; in fact, they appear to be. 2 Criticism to the Twin's Paradox

Translated in a more popular version, formulated by underlines a limitation of the : in 1911 [2], these statements gave rise to points of view are symmetrical only for inertial the “twin’s paradox”. In short: one of two twins remains reference systems. (…) Since there is no symmetry, on Earth while the other gets on a rocket, travels with a Special Relativity is not contradicted by the velocity near to and gets back; since the theory realization that the twin who left Earth is younger consistently sustains that a moving clock runs slower than that his sibling at the time of their reunion.” a ‘stationary’ one, 𝑐𝑐the voyager returns younger than his But this, in fact, does not settle the question. Further, twin, when they meet again. The well-known paradox the “peculiar consequence” became a quite serious issue. comes from the fact that, also consistently, SR takes all The debate on it may not be peaceful and even ignite inertial frames to be equivalent. So, from the point of view passions. For instance, in the 1980s, a trio of authors of the travelling twin it is the other who is moving and the argued with acrimony about whether Einstein changed or discrepancy on ages should be reverted. This is, of course, not his mind about it. In [6], Mendel Sachs sustains that an absurd. “Einstein abandoned his earlier view that there are The French philosopher Henri Bergson caught the material consequences, such as asymmetric aging, essence of this nonsense in his bold criticism to Einstein’s implied in the space-time transformations of relativity SR theory, particularly in the twin’s issue. He “argued theory.” In [7], four years later, the authors present a that there would be no such effect [time passing at “reply to a misleading paper by M. Sachs entitled (…)” different rates for twins] because the Lorentz where “he tried to convince the reader that Einstein transformations performed on the two reference frames changed his mind (…). Also, Sachs insinuates that he were reciprocal” [3]. He was right, though in fact he presented several years ago “convincing mathematical missed the point. arguments” proving that the does not Since Langevin first attempt, many thinkers sought a predict asymmetrical aging in the TP.” The vised author way to explain or ‘undo’ this paradox. “ responds briefly to this criticism in [8]. (1911, 1913) elaborated on Langevin's explanation. Using Now, as we saw above, an “alternative” to the nuisance 's spacetime formalism, Laue went on is to dismiss the paradox as a problem of SR. For instance, to demonstrate that the world lines of the inertially Øyvind Grøn puts it this way [9]: moving bodies maximize the elapsed between “One may get rid of the twin’s paradox at once by two events. He also wrote that the asymmetric aging is noting that in order to be able to meet, depart and completely accounted for by the fact that the astronaut meet again, at least one of the twins must accelerate. twin travels in two separate frames, while the Earth twin And within the special theory of relativity the remains in one frame, and the time of can be principle of relativity is not valid for accelerated made arbitrarily small compared with the time of inertial motion. Acceleration is absolute. Hence, at least one motion. Eventually, Lord Halsbury and others removed of the twins is not allowed to consider himself at rest. any acceleration by introducing the "three-brother" The twin with the greatest average velocity between approach” [4]. None of these arguments are fully the events P1 and P2 is youngest when the twins convincing, so one may find numerous posterior meet at P2”. “explanations” to the paradox, based whether on symmetry break concerning the situation of the twins due The author uses the twin’s paradox as: “a pedagogical to (G-forces, etc.), whether on Doppler entrance to the general theory of relativity. And the above effect or on other factors outside uniform movement. resolution of the twin’s paradox is contrary to the spirit of Is Einstein’s “peculiar consequence” paradoxical? In to the general theory of relativity”, he writes. This [5], Jean-Pierre Luminet sustains the answer is “no”. He appears to be a bizarre assumption. writes: The author of [10] review, for his part, qualifies the “In scientific usage, a paradox refers to results paradox as a “naive interpretation”, affirming that: which are contradictory, i.e. logically impossible. “The really strange thing about time dilation is that But the twin’s paradox is not a logical contradiction, it is symmetrical: if you and I have relative motion, and neither Einstein nor Langevin considered such a then I see your clock to be running (with respect to result to be paradoxical. Einstein only called it our fames), and you see mine to be running slow. “peculiar”, while Langevin explained the different This is just one example of the weird logic of aging rates as follows: “Only the traveler has Einstein's theory of Special Relativity. (...) The naive undergone an acceleration that changed the interpretation – the reason why the situation is called direction of his velocity”. He showed that, of all the a paradox – is to assume that the situation is wordlines joining two events (in this example the completely symmetrical.” spaceship’s departure and return to Earth), the one Quite recently, Gerrit Coddens, the nonconformist or that is not accelerated takes the longest proper time. disillusioned author of [11], reflects on the “protocol The twin’s paradox, also called Langevin effect, which defines the journey (…) with respect to a given

Universal Journal of Physics and Application 15(1): 1-7, 2021 3

frame”, stating that: “It is this selection of a special Consider an observer in a point , the origin of a reference frame which introduces the asymmetry. Hence, supposed immobile coordinates frame . Another 𝟎𝟎 the reference frame wherein we define the protocol for the observer, , in the origin𝐀𝐀 of an inertial𝐎𝐎 frame , is journey will act like an absolute frame and it is this moving with velocity along the -axis, in𝑆𝑆 the positive unavoidable introduction which breaks symmetry between sense. According𝐀𝐀′ to Lorentz transformations, 𝑆𝑆′ 𝑣𝑣 𝑥𝑥 the twins.” In section 4.2, “The true solution of the = ( + / ) paradox”, the author calls into question human “physical = ( + ) 2 intuition” and finally concludes that “there is thus not just where = . (1) 𝑡𝑡 = 𝛾𝛾 𝑡𝑡′ 𝑥𝑥′𝑣𝑣 𝑐𝑐 one twin’s paradox, but an uncountable infinity of them.” ⎧ 1 𝑥𝑥 = 𝛾𝛾 ,𝑥𝑥′ 𝑣𝑣𝑣𝑣′ 2 However, it seems to me that he does not really solve the 𝛾𝛾 �1−𝛽𝛽 ⎨𝑦𝑦 𝑦𝑦′ problem, instead complicates it a lot. At the instant = = 0 a light beam is emitted in ⎩𝑧𝑧 𝑧𝑧′ Finally, one may find on YouTube several videos on the sense of the -axis in both frames, and ; it is 0 0 the subject, such as these two, which justifiably restrict reflected in a mirror𝑡𝑡 placed𝑡𝑡′ at a above (event 1) the analysis to SR: one essentially skeptical [12], and gets back to the𝑦𝑦 observers (event 2). Naturally𝑆𝑆 𝑆𝑆 in′ both understanding the importance of the “turn around” point, frames we have a similar own experience𝑙𝑙, which may be the other presenting a solution by a scientist quite described by the events in , being always null: confident in himself [13]. There are some fine analyses in = / = 2 = 2 / YouTube videos, like these ones; but, along each 𝑆𝑆′ 𝑧𝑧′ = 0 and = 0 (2) explanation, something becomes obscure and unsatisfying. 1 2 1 𝑡𝑡′ =𝑙𝑙 𝑐𝑐 𝑡𝑡′ = 0𝑡𝑡′ 𝑙𝑙 𝑐𝑐 So, there is no explanation at all! �𝑥𝑥′1 �𝑥𝑥′2 To resume, answers based on acceleration and Now, seen 𝑦𝑦from′1 𝑙𝑙 , these coordinates𝑦𝑦′2 become deceleration are tempting ones, but incorrect because the problem directly arises from SR inertial frames; so, we = 𝑆𝑆 = = 2 should not look for an explanation outside these frames. 𝑡𝑡′1 𝑡𝑡′2 = 2 and 2 = =2 2 1 (3) Even if we do so, a supposed ‘resolution’ quickly shows 𝑡𝑡1 �1−𝛽𝛽 𝑡𝑡 �1−𝛽𝛽 𝑡𝑡 = = = = 0. itself as inconsistent. Besides, Special Relativity is, in my �𝑥𝑥1 𝑣𝑣𝑡𝑡1 �𝑥𝑥2 𝑣𝑣𝑡𝑡2 𝑥𝑥1 understanding, the most fundamental theory about space 1 1 2 2 The first𝑦𝑦 equations𝑦𝑦′ 𝑙𝑙 on both𝑦𝑦 sets𝑦𝑦 ′express the time and time structure, energy, etc. It is one of the best dilation phenomenon, from a moving frame towards verified theories in physics [5]; it is compatible with the ‘immobile’ frame – this is, concerning a moving quantum theory, in Dirac’s form – remark that, in his clock towards an ‘immobile’ one: 𝑆𝑆′ founding article, Einstein was careful to point out the 𝑆𝑆 coherence of his theory with Planck's equation: “It is = (4) Δ𝑡𝑡′ noteworthy that the energy and the frequency of a light 2 complex vary with the observer state of motion according This, along with lengthΔ𝑡𝑡 contraction�1−𝛽𝛽 , to the same law.” [1] §8 – and it allows inclusively to deal with accelerated frames and even tachyonic ones, = 1 , (5) according to the PtR theory I presented some years ago, 2 which introduced the reality of negative time flux [14]. So, can be easily deducedΔ directly𝑥𝑥 Δ𝑥𝑥 ′from� − the𝛽𝛽 table (1). Both are it is in SR theory that we must find an answer, but not “on fundamental and unattackable results of Special demand”. And there we find it, indeed! Relativity. It is quite reasonable to assume that there cannot exist This “thought experiment” corresponds exactly to paradoxes in Nature; in this case, that there cannot be a sending a light beam with an angle , given by different aging from twins. But then, how is it, despite the sin = 1 (Figure 1); the beam is reflected in a 𝜑𝜑 reality of time dilation? The real answer to the paradox is mirror at a distance2 placed on the perpendicular to the to be found in the profound understanding of break of -axis𝜑𝜑 from� −the𝛽𝛽 point with coordinate = , and then returned to a 𝑙𝑙point also in this axis, with simultaneity and of spacetime behavior, this is, of our 𝟏𝟏 1 1 𝑥𝑥coordinate = 2 =𝐎𝐎 [see Figure 1]. 𝑥𝑥It results𝑣𝑣𝑡𝑡 that, ways to measure phenomena in it. In the end, we discover 𝟐𝟐 in fact, the light beam takes𝐎𝐎 a longer time to reach the that Special Relativity is yet a wonderful box full of 2 1 2 surprises (as well as Nature, which the theory translates) point than𝑥𝑥 it does𝑥𝑥 in𝑣𝑣 𝑡𝑡the perpendicular way (equal to and that our difficulties arise from approaches based, after = 2 / ): exactly = / 1 . 𝐎𝐎𝟐𝟐 all, on ‘common sense’. On the other hand, the “thought2 experiment” 2 2 2 𝑡𝑡concerning′ 𝑙𝑙 𝑐𝑐 a beam of𝑡𝑡 light𝑡𝑡′ emitted� − 𝛽𝛽 in the sense of the -axis clearly shows the phenomenon, 2. The Basics: Time Dilation and but it also displays another crucial feature of Special Length Contraction Relativity:𝑥𝑥 the break of simultaneity.

4 Criticism to the Twin's Paradox

Figure 1. The experience of the light beam emitted in the perpendicular to a moving -axis.

𝑦𝑦′ 3. The Twin’s Paradox of SR theory, we must search for an error somewhere in the premises. And, in fact, there is one, at least: it lies in 3.1. The One-Way Voyage the premise “at the same moment (…)”. The error comes from not taking the break of simultaneity into account. We We must get rid of rockets, accelerations and the idea must keep in mind that Lorentz transformations either for of high velocities, because this has nothing to do with the coordinate, either for coordinate, are functions of reality of time dilation and Einstein’s “peculiar both the correspondent coordinates in the other frame. consequence”. Instead, we will extensively use the This𝑥𝑥 is quite easy to understa𝑡𝑡 nd in the case of the length reflection experiment to measure time, this is, as a clock. coordinate (it is equivalent to the ) But, instead of thinking the problem based on the but harder in what comes to the time coordinate: though in symmetry of the situation for the movement of in the frame the measure of time does not depend on and of in , we will analyse it otherwise. Take a third the position of a given point in space, the 𝑃𝑃 observer, , immobile at the point , and, 𝐀𝐀for′ our𝑆𝑆 correspondent𝑆𝑆 time depends on𝑡𝑡 and on ; we purposes,𝐀𝐀 consider𝑆𝑆′ that they are all twins. Consider also must understand that, for instance, the𝐏𝐏 aforementioned 𝟐𝟐 𝑃𝑃 𝑃𝑃 𝑃𝑃 [following 𝐁𝐁the usual line of thought] that𝐎𝐎 sends a light coordinates and 𝑡𝑡′ are the coordinates𝑡𝑡 𝑥𝑥of the beam at the same moment the observer does it; the observer in each frame, and respectively: mirror above is again at the coordinate𝐁𝐁 = = . 𝑡𝑡2 𝑡𝑡′2 = and = The beam reaches the mirror, is reflected𝐀𝐀′ there and 𝐀𝐀′ 𝑆𝑆 𝑆𝑆′ = 1 . = ′ = and = 0′ reaches the point𝐁𝐁 at the instant 𝑦𝑦 𝑙𝑙 𝑙𝑙′ 2 𝐴𝐴 2 2 𝐴𝐴 2 2 𝑡𝑡 𝑡𝑡 ′ 𝑡𝑡′ 𝑡𝑡′ 2𝐴𝐴′ 2 � This2 is𝐴𝐴 2quite 2different from2𝐴𝐴′ the coordinates⇒ 𝑡𝑡′ in𝑡𝑡 � −of𝛽𝛽 𝟐𝟐 2 2 𝑥𝑥 𝑥𝑥 𝑣𝑣𝑡𝑡 𝑥𝑥′ 𝐎𝐎 = = = , at the instant : 𝑆𝑆′ 𝐀𝐀 𝑙𝑙 𝑙𝑙′ 2 that is, before the 𝑡𝑡arrival𝐵𝐵2 of and𝑡𝑡 ′of2 his own reflected 𝑡𝑡 = beam, which occurs simultaneously𝑐𝑐 𝑐𝑐 at time = 1 2 𝐴𝐴2 𝑡𝑡 𝐀𝐀′ ⎧𝑡𝑡′ 2 / 1 . This is, of course, an obvious result. ⎪ = . 𝑡𝑡2 � − 𝛽𝛽 But consider2 now the experiment from the point of 1 2 2 𝐴𝐴2 𝑥𝑥 𝑡𝑡′ � − 𝛽𝛽 ⎨𝑥𝑥′ − 2 view of : the observer is moving along the -axis We will return to⎪ this issue in a while. For now, using ⎩ � − 𝛽𝛽 with velocity . Time is dilated to suitable notations, we will write the inverse Lorentz 𝐀𝐀′ 𝐁𝐁 𝑥𝑥 transformation, for the same time coordinate in , −𝑣𝑣 𝑡𝑡2𝐵𝐵 = = = = , as: 1 𝐵𝐵2 1 2 𝐵𝐵2 𝑡𝑡 𝑡𝑡′ / / 𝑆𝑆 𝑡𝑡′ 2 2 𝐵𝐵= 𝐴𝐴 = = and this means that — symmetrically, which is logical — 𝑡𝑡 𝑡𝑡 𝑡𝑡 2 2 � − 𝛽𝛽 � − 𝛽𝛽 𝑡𝑡 and 𝑡𝑡−𝑥𝑥𝐵𝐵 𝑣𝑣 𝑐𝑐 𝑥𝑥𝐵𝐵 𝑣𝑣 𝑐𝑐 (6) 𝐴𝐴 2 𝐵𝐵 2 𝐴𝐴 2 receives his own beam before meeting ! So, the laps 𝑡𝑡′ = �1−𝛽𝛽 𝑡𝑡′ = �1−𝛽𝛽 = 𝑡𝑡′+− �1−𝛽𝛽. of time between each observer receiving his own reflected � 𝑣𝑣𝑣𝑣 � 𝑥𝑥𝐵𝐵 −𝑣𝑣𝑣𝑣 𝑥𝑥𝐵𝐵 𝐴𝐴 2 𝐵𝐵 2 𝐴𝐴 2 𝐀𝐀beam′ and the other observer’s doing so is𝐁𝐁 opposite for 𝑥𝑥This′ clearly− �1−𝛽𝛽 shows that𝑥𝑥 ′ the �synchronous1−𝛽𝛽 𝑥𝑥′ �clocks1−𝛽𝛽 in , each frame. This appears to be an insolvable contradiction ‘owned’ by and , appear as asynchronous in the and is equivalent to the traditional twin’s paradox. frame of the observer . 𝑆𝑆 If we strongly believe in the consistency of Nature and Now, if we𝐀𝐀 look 𝐁𝐁for the coordinates in of the 𝑆𝑆′ 𝐀𝐀′ 𝑆𝑆′ Universal Journal of Physics and Application 15(1): 1-7, 2021 5

observer corresponding to the instant = 0 where the movement of in the same velocity the light beam is emitted by the observer , we get from 0 | | the transformation𝐁𝐁 table above, considering𝑡𝑡′ that 𝐁𝐁= 𝑆𝑆′ = = . = = for every instant , 𝐀𝐀′ Δ𝑥𝑥′ Δ𝑥𝑥 𝑣𝑣′𝐵𝐵 − − −𝑣𝑣 𝐵𝐵 2 2 = 3.2. The Simultaneous ReceptionΔ𝑡𝑡′ Δ 𝑡𝑡 𝑥𝑥 𝑥𝑥 𝑣𝑣𝑡𝑡 2 𝑡𝑡 𝛽𝛽 𝑡𝑡2 (7) 2 𝑡𝑡′𝐵𝐵0 = − �1−𝛽𝛽. If we wish to evaluate the relative aging of the twins, � 𝑥𝑥2 we may do it in an alternative way at the very moment 𝐵𝐵0 2 Remark that, at th𝑥𝑥is′ moment,�1−𝛽𝛽 for the observer is they meet, because only then both clocks are in mutual in the past. For the instant , when receives its beam proximity. In the frame , the observer and his and meets , we get 𝐀𝐀′ 𝑩𝑩 = / 1 2 reflected light beam meet in time ; 𝑡𝑡 𝐀𝐀′ on the other hand, the beam𝑆𝑆 experience made 𝐀𝐀by′ takes2 = = 1 = 2 2 𝐁𝐁 2 a time . So, for both light𝐁𝐁 beams (the𝑡𝑡 one𝑡𝑡′ sent� by− 𝛽𝛽 1−𝛽𝛽 2 (8) 𝐵𝐵2 2 2 2 2 and the one sent by ) to arrive at the same time𝐁𝐁 to the = �1−𝛽𝛽= 0. 𝑡𝑡′ 𝑡𝑡 𝑡𝑡 � − 𝛽𝛽 𝑡𝑡′ 𝑡𝑡′2 𝐀𝐀′ � 𝑥𝑥2−𝑣𝑣𝑡𝑡2 point , must send his own beam at the instant 𝐵𝐵2 2 𝐁𝐁 The first result𝑥𝑥′ represents�1−𝛽𝛽 the time dilation of ; but, 𝟐𝟐 = = 1 1 . above all, it means that, from the point of view of , 𝐎𝐎 𝐁𝐁 𝑡𝑡2 2 when he reaches , both have aged exactly the same: . In the frame𝑡𝑡3 𝑡𝑡2 −, 𝑡𝑡′the2 inverse𝑡𝑡2 � − �transformation− 𝛽𝛽 � gives So, as transcribed in coordinates, the difference𝐀𝐀′ 2 (making = ): between final and𝐁𝐁 initial length and time coordinates 𝑡𝑡for′ 𝑆𝑆′ 𝑩𝑩 𝑥𝑥3 𝑥𝑥2 1 1 the experience “ sends and receives a reflected light = 3 = 2 2 = 2 = 3 𝑣𝑣𝑥𝑥2 2 𝑣𝑣𝑥𝑥2 2 𝑣𝑣𝑥𝑥2 ′ 𝑡𝑡 1− 𝑡𝑡 � − �1− 𝛽𝛽 � − 𝑡𝑡 1− ′ beam” yields: ⎧ 3 𝑐𝑐 𝑐𝑐 𝑐𝑐 2 2 2 𝐀𝐀′ ⎪𝑡𝑡 2 + 2 1 2 − 𝑡𝑡 𝑡𝑡 − 𝑡𝑡 = � − 𝛽𝛽 = � − 𝛽𝛽 = � − 𝛽𝛽 + = + . 2 = = + = 31 3 2 2 1 2 21 2 2 2 ⎨ 3 𝑥𝑥 − 𝑣𝑣𝑡𝑡 𝑥𝑥 − 𝑣𝑣𝑡𝑡 𝑣𝑣𝑡𝑡 � − 𝛽𝛽 𝑥𝑥 − 𝑣𝑣𝑡𝑡 2 2 2 1−𝛽𝛽 𝛽𝛽 𝑡𝑡2 (9) ⎪𝑥𝑥′ 2 2 2 𝑣𝑣𝑡𝑡 𝑥𝑥′ 𝑣𝑣𝑡𝑡 𝐵𝐵2 𝐵𝐵2 0𝐵𝐵 2 2 2 2 ⎩ � − 𝛽𝛽 � − 𝛽𝛽 � − 𝛽𝛽 Δ𝑡𝑡′ = 𝑡𝑡′ − 𝑡𝑡=′ 𝑡𝑡 ��. 1−𝛽𝛽 �1−𝛽𝛽 � �1−𝛽𝛽 But, since = , it results = 0 and, therefore, � 𝑥𝑥2 2 2 2 = = 2 Δ𝑥𝑥′𝐵𝐵2 −𝑥𝑥′𝐵𝐵0 − �1−𝛽𝛽 𝑥𝑥 𝑣𝑣𝑡𝑡 𝑥𝑥′ Both differences express a dilation (in particular, = = . > ); but in the end, coherently, their quotient 𝑡𝑡′3 −𝑡𝑡3 𝑡𝑡′2 − 𝑡𝑡2 � gives the velocity of relatively to : Remark that a 𝑥𝑥negative′3 𝑣𝑣𝑡𝑡2 time𝑥𝑥2 appears once again, Δ𝑡𝑡′2𝐵𝐵 𝑡𝑡2 precisely the symmetrical of . This is the key to 𝐁𝐁 𝐀𝐀′ understand and resolve the problem. It means that the = = = . 3 𝐵𝐵2 2 emission of the beam by , which𝑡𝑡 in the frame is 𝐵𝐵 Δ𝑥𝑥′ 𝑥𝑥 Further, 𝑣𝑣′ 𝐵𝐵2 − 2 −𝑣𝑣 posterior to the emission of the beam by , is in this Δ𝑡𝑡′ 𝑡𝑡 𝐁𝐁 𝑆𝑆 = = one’s proper frame anterior to it. So, one concludes that, in , is younger when he sends his𝐀𝐀 ′light beam = = , 𝑆𝑆′ Δ𝑡𝑡′𝐵𝐵2 Δ𝑡𝑡′𝐴𝐴2 𝑡𝑡′𝐴𝐴2 than is when he sends his own beam. Further: when � 𝑆𝑆′ 𝐁𝐁 and this makes allΔ 𝑥𝑥sense.′𝐵𝐵2 TheΔ𝑥𝑥′𝐴𝐴 2mutual𝑥𝑥′𝐴𝐴 2dilation may seem the moving observer reaches and both receive strange and contradict Relativity. But it does not. First, it their 𝐀𝐀reflected′ beams, a time laps has passed in . is to be expected regarding time; second, concerning , Therefore, in this frame,𝐁𝐁 a total time𝐀𝐀 ′laps exist since 2 it comes from the fact that the phenomenon of length the emission of the beam by , which𝑡𝑡′ is given by 𝑆𝑆′ contraction requires the measure of to be taken atΔ the𝑥𝑥′ Δ𝑡𝑡′ = = , same instant ; this does not happen here (where 𝐁𝐁 Δ𝑥𝑥′ simply expresses a difference of coordinates). this being exactly the time2 required3 2in for the beam 𝑡𝑡′ Δ𝑥𝑥′ Δ𝑡𝑡′ 𝑡𝑡′ − 𝑡𝑡′ 𝑡𝑡 Also remark that all the precedent reasoning may be emitted by to return to him and to reach ! 𝑥𝑥′ applied as well to the movement of relatively to , Remember that this describes how observes,𝑆𝑆 in his for this one emitting the light beam; it is enough to proper frame 𝐀𝐀′ , the experience “emission and reception𝐁𝐁 𝐀𝐀′ 𝐁𝐁 consider the frame centred in and the -axis of a light beam by his twin ”. So, in𝐀𝐀′ a way, this result oriented in the opposite sense. was to be expected:𝑆𝑆′ it is the symmetrical process and 𝑆𝑆 𝐎𝐎𝟐𝟐 𝑥𝑥 Finally, from the strict point of view of the observer , corresponds to time dilation;𝐁𝐁 but it becomes fully the path between and may be seen as a rigid rule, understandable by the fact that in the frame the 𝐀𝐀′ immobile in the frame , with length = . Therefore, observer emits his beam before his twin does it. 𝐀𝐀 𝐁𝐁 it appears in with a contracted length | | = The point here is that this reasoning is interchangeable𝑆𝑆′ 𝑆𝑆 Δ𝑥𝑥 𝑥𝑥2 1 , which covers in the same time between frames,𝐁𝐁 simply permuting and𝐀𝐀 ′ : for each 𝑆𝑆′ Δ𝑥𝑥′ = =2 1 it takes the light to go and return twin, the other sends his light beam before he does it! This 𝑥𝑥2� − 𝛽𝛽 𝐀𝐀′ from the mirror above2 him; so, also coherently, we get for is a quite amazing conclusion but 𝐀𝐀is′ the basis𝐁𝐁 for the Δ𝑡𝑡′ 𝑡𝑡′2 𝑡𝑡2� − 𝛽𝛽

6 Criticism to the Twin's Paradox

understanding of how Nature resolves the imbroglio. The So, we will have (almost) similar equations to (7) and same delay compensates the longer time laps = (8) for = = 0 and for the final = : for the moving twin’s experience compared with the time 2 = 4 2 laps for the immobile one’s experience, in sortΔ 𝑡𝑡that,′ 𝑡𝑡at 𝐭𝐭′ 𝐭𝐭 2 = 𝐭𝐭 1 𝑡𝑡 = 𝛽𝛽 𝑡𝑡2 and (12) the end, both twins have aged exactly the same: their 2 2 2 𝐴𝐴0 = �1−𝛽𝛽 = 0. 𝑡𝑡′ 𝐭𝐭′ − 𝐭𝐭′𝐴𝐴4 𝑡𝑡2� − 𝛽𝛽 𝑡𝑡′2 proper time laps . � 𝑥𝑥2 � 2 𝐴𝐴4 We see then that there is indeed no real conflict In𝐱𝐱′𝐴𝐴 each0 − case,�1− 𝛽𝛽we are interested𝐱𝐱′ in the summative time 𝑡𝑡′2 between frames evaluations. The mutual time dilation ( ) and length coordinates, from the beginning (the departure is a real relativistic phenomenon but does not affect the of from ), symbolized by the bold capital letters 𝑡𝑡2 effective aging of and . and . To obtain them we must add to and 𝟎𝟎 the 𝐀𝐀respective′ 𝐎𝐎 new initial coordinates given by (10). The 𝐀𝐀′ 𝐁𝐁 𝐴𝐴 𝐴𝐴 𝐴𝐴 𝐴𝐴 3.3. The Round Trip 𝐓𝐓result′ is: 𝐗𝐗′ 𝐭𝐭′ 𝐱𝐱′ This seems to be true (and to settle the question) = 1 = (13) concerning and . But what about ? = 2 𝐓𝐓′𝐴𝐴0 𝑡𝑡2� − 𝛽𝛽 𝑡𝑡′2 Well, in the frame , the age of the twins and is 2𝑥𝑥2 � 2 supposed to𝐀𝐀 ′be always𝐁𝐁 the same. So,𝐀𝐀 a reasonable and 𝐗𝐗′𝐴𝐴0 − �1−𝛽𝛽 hypothesis is that, when𝑆𝑆 reaching the starting𝐀𝐀 point, after𝐁𝐁 reverting his movement in , and should meet = 1 + 2 1 (14) with the same age. But how is this possible? The answer 2 𝟐𝟐 𝐴𝐴4 = 2 . �1−𝛽𝛽 is to be found in a subtle and𝐎𝐎 intriguing𝐀𝐀′ interference𝐀𝐀 that 𝐓𝐓′ 𝑡𝑡 �� − 𝛽𝛽 � � 𝑥𝑥2 2 comes out in the turning point . First, we see 𝐗𝐗from′𝐴𝐴4 − �1−=𝛽𝛽 , that, for simply Let us re-examine the issue from the point of view of 𝟐𝟐 reversing its course, at that moment the advance in time of 𝐎𝐎 𝐴𝐴0 2 the observer . As we have seen, when he reaches , is overturned: both ‘find’𝐓𝐓′ themselves𝑡𝑡′ with the𝐀𝐀 same′ age. the coordinates of are This is the most astounding phenomenon! 𝐀𝐀′ 𝐎𝐎𝟐𝟐 𝐀𝐀 Then, at the end of the double light beam experience – 𝐀𝐀 = 𝑡𝑡2 (10) this is, the round trip –, meets after a time lapse 2 𝑡𝑡′𝐴𝐴2 = �1−𝛽𝛽 . , which is equal for both. Remark that, in fact, the � 𝑥𝑥2 equation includes both contraction𝐀𝐀′ and𝐀𝐀 dilation positive 𝐴𝐴2 2 𝐴𝐴4 Take a close look at𝑥𝑥 ′these −formulae.�1−𝛽𝛽 One must remind components𝐓𝐓′ and that all the reasoning leads to the same that they concern the beam experience made by , not by result if it is applied to the movement of in relation to his twin (where the coordinate would be equal to . So, there is no longer a contradiction regarding the 𝐀𝐀′ and to ). What the formulae say is essentially measure of time in the two reference𝐀𝐀 ′frames; both 𝐀𝐀 𝑡𝑡2 that is in the future for when this one, after a time twins𝐀𝐀 have aged the same: = = . 𝑡𝑡′2 𝑡𝑡′2𝐴𝐴 𝑡𝑡2 laps , reaches ; the temporal difference is given by One may also express as ′ 𝐀𝐀 𝐀𝐀′ 𝐓𝐓4 𝐓𝐓𝐴𝐴 4 𝐓𝐓′𝐴𝐴4 𝑡𝑡′2 𝐁𝐁 2 4 2 = = ; = 𝐓𝐓 or = . 1 2 2 2 𝛽𝛽 1 1 𝐴𝐴2 2 2 2 4 − 𝛽𝛽 2 4 − 𝛽𝛽 2 [for instance,Δ for𝑡𝑡′ 𝑡𝑡′= 0−.5𝑡𝑡,′ this gives 𝑡𝑡′ = 1/3 ]. In 𝐓𝐓 2 𝑡𝑡 𝐓𝐓 2 𝑡𝑡′ − 𝛽𝛽 If = 0, this conduces to = and the consistent this future moment, is at a distance | | behind . � − 𝛽𝛽 − 𝛽𝛽 result = 2 . For > 0, > 2 (for instance, This is compatible𝛽𝛽 with / = Δ 𝑡𝑡but′ is strange!𝑡𝑡′ 2 2 𝐴𝐴2 making𝛽𝛽 = 0.5 , it comes 𝑡𝑡 =𝑡𝑡′ 7/3 ). In fact, However, stranger things𝐀𝐀 are yet to come.𝑥𝑥 ′ 𝐀𝐀′ 4 2 4 2 𝐴𝐴2 𝐴𝐴2 lim T 𝐓𝐓= ∞ for𝑡𝑡 ′ 1. 𝛽𝛽Besides𝐓𝐓, the difference𝑡𝑡′ between From a theoretical point𝑥𝑥′ of 𝑡𝑡′view, −when𝑣𝑣 instantly 4 2 the summative𝛽𝛽 time for moving𝐓𝐓 and immobile𝑡𝑡′ frames is reverts his movement, it is advisable to consider a new 4 given by 𝛽𝛽 → immobile frame , the former reset and 𝐀𝐀ce′ ntered in

. Then, and his also new frame will move in the 2 = = = , 2 2 opposite direction𝑺𝑺 of the -axis,𝑆𝑆 with velocity , 1 1 2 𝟐𝟐 ′ 𝛽𝛽 ′ 𝛽𝛽 𝑡𝑡 ′ 𝐎𝐎 𝐀𝐀′ 𝑺𝑺′ 4 2 2 2 𝐴𝐴0 being coincident with at the instant = = 0. In this is,𝐓𝐓 in− modulus,𝑡𝑡 the non-𝑡𝑡summative time2 coordinate−𝐭𝐭 of 𝑥𝑥 −𝑣𝑣 𝑺𝑺′ − 𝛽𝛽 � − 𝛽𝛽 these circumstances, naturally, Lorentz transformations relatively to at the turning point. One should say 𝑺𝑺 𝐭𝐭′ 𝐭𝐭 apply by simply reversing the sign of . that this increment in time is provided by the reversion of As we did before, for , we will write the𝐀𝐀 movement of𝐀𝐀 ′ . In a way, it is what is needed to 𝑣𝑣 bring to the present of . 𝐁𝐁 / / = = To end this paper𝐀𝐀′ in beauty, one just needs to deduce 2 2 𝐭𝐭+𝐱𝐱𝐴𝐴 𝑣𝑣 𝑐𝑐 𝐭𝐭−𝑥𝑥2 𝑣𝑣 𝑐𝑐 (11) the summative𝑨𝑨 coordinates𝐀𝐀 for′ the observer . His initial 𝐴𝐴 2 2 𝐭𝐭′ = �1−𝛽𝛽 = �1−.𝛽𝛽 coordinates are those expressed in (8): � 𝐱𝐱𝐴𝐴 +𝑣𝑣𝐭𝐭 𝑣𝑣𝐭𝐭−𝑥𝑥2 2 2 𝐁𝐁 𝐱𝐱′𝐴𝐴 �1−𝛽𝛽 �1−𝛽𝛽

Universal Journal of Physics and Application 15(1): 1-7, 2021 7

1 younger than the one who "stayed at home": they meet = = 1 2 again at exactly the same age. 1 2 𝐵𝐵2 2 − 𝛽𝛽 2 ⎧𝑡𝑡′ 𝑡𝑡 2 𝑡𝑡 � − 𝛽𝛽 ⎪ = � − 𝛽𝛽= 0. 21 2 ⎨ 𝐵𝐵2 𝑥𝑥 − 𝑣𝑣𝑡𝑡 𝑥𝑥′ 2 Regarding⎪ the new frames and , since = 0 at ⎩ � − 𝛽𝛽 REFERENCES every moment, one gets: 𝑺𝑺 𝑺𝑺′ 𝐱𝐱𝐵𝐵 [1] Hendrik A. Lorentz. Albert Einstein e Hermann Minkowski. = The principle of relativity - Portuguese translation. Calouste 𝐭𝐭 (15) Gulbenkian Foundation, Lisbon, 3rd edition. 1983; 55-62, 2 𝐭𝐭′𝐵𝐵 = �1−𝛽𝛽 ; §4. The translation from Portuguese to English is mine. I 𝑣𝑣𝐭𝐭 also replaced for the modern notation . An alternative � 2 and this gives, for the turning𝐱𝐱′𝐵𝐵 point,�1−𝛽𝛽 translation may be found in Doc. 23, p. 153 of https://einsteinpapers.press.princeton.edu/vol2𝑉𝑉 𝑐𝑐 -trans/167 = 0

= 0 , [2] Paul Langevin. L’évolution de l’espace et du temps, Scientia 𝐭𝐭′𝐵𝐵0 X: 31-54. which is quite natural, meaning� 𝐵𝐵0 that is not affected by 𝐱𝐱′ [3] Ray Scott Percival, note on Challenger to Einstein’s theory the reversion of the movement of ; and also 𝐁𝐁 of time. The New Bergson – Duration and Simultaneity, Higher, October 6, 2000; accessible in: = 𝐀𝐀′ https://philpapers.org/archive/PERBCT.pdf 1 2 𝐵𝐵4 𝑡𝑡 ⎧𝐭𝐭′ 2 [4] Twin’s paradox. Wikipedia; accessible in: https://en.wikipe ⎪ = � − 𝛽𝛽 . dia.org/wiki/Twin_paradox 1 2 𝐵𝐵4 𝑥𝑥 ⎨𝐱𝐱′ 2 [5] Jean-Pierre Luminet. Time, Topology and the Twin’s Therefore, one ⎪obtains the following summative ⎩ � − 𝛽𝛽 paradox; accessible in: https://arxiv.org/ftp/arxiv/papers/09 coordinates: 10/0910.5847.pdf = (16) [6] Mendel Sachs. On Einstein’s later view of the twin’s = 0, paradox, Foundations of Physics 15 (9):977-980, 1985; 𝐓𝐓′𝐵𝐵0 𝑡𝑡′2 and � abstract accessible in: https://philpapers.org/rec/SACOEL 𝐗𝐗′𝐵𝐵0 [7] Waldyr A. Rodrigues & Marcio A. F. Rosa. The meaning of = 1 + time in the theory of relativity and “Einstein’s later view of 2 1 (17) 𝐵𝐵4 2 2 the Twin’s paradox”, Foundations of Physics 19 (6):705-724, 𝐓𝐓′ = 𝑡𝑡 �� . − 𝛽𝛽 �1−𝛽𝛽 � 1989; abstract accessible in: https://philpapers.org/rec/ROD � 𝑥𝑥2 2 TMO-2 This finally means𝐗𝐗′𝐵𝐵4 that�1− 𝛽𝛽 ages exactly the same from his twins and , when these ones meet again, which [8] Mendel Sachs. Response to Rodrigues Rosa on the twin’s is , thus confirming our hypothesis.𝐁𝐁 paradox, Foundations of Physics 19 (12):1525-1528, 1989; 𝐀𝐀′ 𝐀𝐀 abstract accessible in: https://philpapers.org/rec/SACRTR 4 𝐓𝐓 [9] Øyvind Grøn, The twin’s paradox and the principle of relativity; accessible in: https://arxiv.org/ftp/arxiv/papers/10 4. Conclusions 02/1002.4154.pdf Nature creates the effect of temporal dilation; but it also [10] The twin’s paradox: Is the symmetry of time dilation provides ways to overcome it, under certain circumstances, paradoxical? The University of New South Wales, Sidney, avoiding contradictions and ensuring that physical events Australia; accessible in: http://newt.phys.unsw.edu.au/einst einlight/jw/module4_twin_paradox.htm are independent of coordinate frames. The main element in this ‘magic’ is synchronicity, including the fact that the [11] Gerrit Coddens. What is the reason for the asymmetry distinction between past and future is not an objective between the twins in the twin’s paradox?. feature: it depends on coordinate frames. 2019.hal-02263337v5, 2020. Long ago, Einstein demonstrated, in his seminal paper [12] The Twin’s paradox Explained and Resolved, 2012; on Relativity, that synchronicity is broken from one accessible in: https://www.youtube.com/watch?v=K6iMAjf coordinate frame to another. This is a crucial fact. Once z1k8 we deeply understand it, which is quite difficult because it [13] Twin’s paradox: the real explanation (no math) ; accessible often goes against the common sense that – for cause – in: https://www.youtube.com/watch?v=noaGNuQCW8A still rules our mind, we begin to be able to find solutions. “Acceleration isn’t the explanation” The final conclusion here is that, despite the reality of [14] Luís Dias Ferreira. Tachyons and Pseudotachyonic time dilation, it is not true that the "traveling twin" returns Relativity, Concepts of Physics, Vol. IV Nr. 1, 2007.

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