The Solution of the Twin Paradox Consistent with the Time-Energy Relationship Denis Michel

The solution of the twin paradox consistent with the time-energy relationship Denis Michel

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Denis Michel. The solution of the twin paradox consistent with the time-energy relationship. 2015. hal-01182589v1

HAL Id: hal-01182589 https://hal.archives-ouvertes.fr/hal-01182589v1 Preprint submitted on 1 Aug 2015 (v1), last revised 31 Aug 2015 (v2)

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The solution of the twin paradox consistent with the time-energy relationship

Denis Michel

Universite de Rennes1. Campus de Beaulieu Bat. 13. 35042 Rennes France. [email protected]

Abstract troubled by the consequences of his own work. He first explained the experience of half-turn of a clock using The time dilation effect of the round trip, known as the time dilation of special relativity [4]. Later, he at- the twin paradox, has strangely been treated in general tributed the asymmetry between the clocks to a pseudo- relativity in absence of gravity and in special relativity gravitational effect of the about turn [5]. In fact as shown in absence of uniform motion. The alternative Doppler below, the result of this experiment does not require the treatment proposed here reveals a new solution unre- relativity theory and can be found using the classical lated to the Lorentz dilation factor. This solution is Doppler equation. The wrong treatment of the twin ex- then shown consistent with the general time-energy re- periment would be to use special relativity tools (which lationship which applies to all cases of time distortion, hold exclusively for inertial frames) and to arbitrarily de- including special relativity, gravity and cosmological ex- cide who travels and who remains at rest. This can be pansion. readily shown: Twin brothers are separated; one of them remains inert while the other crosses a distance dl before joining his brother. After their reunion, the paths of the twins in Minkowski’s space (ds) are the same, as they 1 Introduction started from the same point and arrived to the same point of spacetime. If one decides that it is the twin Apparent time dilation corresponds to wavelength in- whose proper time is labelled t0 who travels, crease, or equivalently frequency decrease. This phenomenon is particularly obvious for distant supernovae ds2 = (cdt)2 = (cdt0)2 − dl2 (1a) whose large redshift (for example λapp/λ = 1.5), pre- q 0 0 v2 cisely coincides with an increased duration of brightness with v = dl/dt , one obtains dt/dt = 1 − c2 , and (1.5-fold longer in the example) [1]. In special relativity conversely if it is the other twin (of proper time t) who (uniform motions), a time interval corresponds to a given travels, number of periods (say n) so that ∆t0 > ∆t is equivalent ds2 = (cdt)2 − dl2 = (cdt0)2 (1b) 0 0 to nT > nT and to ν < ν. In other words, wave dis- q 0 v2 tortion exactly compensates spacetime distortion in such with v = dl/dt, dt /dt = 1 − c2 . Hence, it is impossi- a way that (i) the speed of light and (ii) the number of ble to break the symmetry. No physical experiment can periods, are both maintained identical for all observers measure the absolute velocity of a frame with a uniform [2]. In addition in the twin paradox, following the idea of motion in which it is conducted, so that one can not as- Darwin [3], there is a difference in the number of periods. sign the speed v specifically to one twin. This problem This phenomenon generates a new differential ageing ef- is at the origin of the term paradox, but now most au- fect irrespective of the Doppler formula used, showing thors point that in fact, the moving twin does not remain that it is not a matter of special relativity. Finally, the in a single inertial frame since his motion is not uniform, time-energy relationship is shown to predict and unify all and certain authors recourse to general relativity to treat the types of time distortion. this situation. Although many interpretations of this ac- tively debated paradox have already been described in the context of either special or general relativity, a sim- 2 The twin paradox pler one is proposed below. To simplify the treatment, let us consider a collinear round trip (starting from a Contrary to special relativity in which all the frames are spatial station to eliminate a role for gravity). When uniformly moving, in the famous twin paradox, the sym- located at a distance D from his sedentary brother, the metry of special relativity is broken by the moving twin, travelling twin makes an about-turn at constant speed even he flies at constant speed. Einstein himself was and infinite acceleration (like a frontal elastic collision).

1 v The twins continuously exchange light pulses with the journey and is shrinked during the 1 − c th of the jour- same fundamental wavelength at the origin. Viewed by ney. Wavelengths can be averaged geometrically over the the travelling twin, things are very simple. He perceives a whole trip as follows. dilated Doppler effect from his resting brother before his about-turn and a shrinked Doppler effect during the trip 2.1.1 Home clock perceived by the traveller back. Viewed by the resting twin, things are less simple because as explained in [3], he cannot perceive the Using the classical Doppler formula, change of Doppler effect during the turn because light 1 app 1 2 2 λ v v 2 v emitted at this point takes a time D/c to reach him, so RT = 1 + 1 − = 1 − (2) that when he perceives it, the travelling twin has already λ c c c2 crossed d = vD/c towards him. The switch between the blue and red shifts does not occur at D but at D + d, or with the relativistic Doppler formula, v D 1 + . This mere fact explains why the number of 1 c s s ! 2 pulses received by the twins is not symmetrical. Hence, λapp 1 + v 1 − v RT = c c = 1 (3) the twin paradox can not help distinguishing between v v λ 1 − c 1 + c the relativistic and classical Doppler effects. The number of wave crests received by the inert twin is lower than and using the conjectural Doppler formula, that received by the travelling twin, which means that 1 app 1 2 − 2 the symmetry is broken contrary to special relativity in λ v v − 2 v RT = 1 + 1 − = 1 − (4) which the number of pulses is the same for all observers. λ c c c2 In the numerous debates on the twin paradox, a fre- quent argument against the followers of general relativity 2.1.2 Traveller’s clock perceived by the resting is that the phase of acceleration or pseudo-gravity can brother be as short as desired compared to the duration of the trip. We see in the mechanism described here that this Using the classical Doppler formula, phase can indeed be reduced to zero in an instantaneous 1 app v v 2 about-turn like an elastic collision, but the time period λ v (1+ c ) v (1− c ) TR = 1 + 1 − of asymmetry remains equal to the fraction v/c of the λ c c (5) trip. v/2c 1 1 + v v2 2 = c 1 − 1 − v c2 2.1 Wavelength geometric averaging c app with the relativistic Doppler formula, λRT is the wavelength of the resting source viewed by app the traveller and λTR is the wavelength of the travel- 1 app v (1+ v ) v (1− v )! 4 ling source viewed by the resting twin. The asymme- λ 1 + c 1 − c TR = c c try between the twins will be evaluated using either the λ 1 − v 1 + v classical, the relativistic, or the conjectural [2] Doppler c c (6) formulas, on the basis that the Doppler effects perceived 1 + v v/2c = c by the resting twin is dilated during the 1 + v th of the v c 1 − c

Table 1: Mean wavelength averaged over the whole round trip, perceived by the resting (R) and travelling (T ) twins and calculated using diﬀerent Doppler formulas. All of them give the same ratio of wavelength exchanged during round trip. For a trip at speed v/c = 0.3, the travelling source aged 10% slower than its resting counterpart.

Point of view Classical Relativistic Conjectural

app λ q 2 q 2 Traveller RT 1 − v 1 1/ 1 − v λ c2 c2 v v v app s v ! c s v ! c s v ! c λTR 1 + c q v2 1 + c 1 + c q v2 Inert v 1 − c2 v v / 1 − c2 λ 1 − c 1 − c 1 − c v v v app s v ! c s v ! c s v ! c hλTR i 1 + c 1 + c 1 + c Ratio app v v v hλRT i 1 − c 1 − c 1 − c

2 and using the conjectural Doppler formula, 2.2 Arithmetic mean frequency

− 1 Apparent temporal paradoxes often disappear if conceiv- app v v 2 λ v (1+ c ) v (1− c ) TR = 1 − 1 + ing time as a frequency depending on the energetic status λ c c of the system. Frequencies provide a very intuitive idea (7) v/2c − 1 1 + v v2 2 of time when applied for example to heart beats, days = c 1 − and seasons. The appropriate tool for averaging both 1 − v c2 c wavelengths and frequencies is the geometric mean, but the traditional arithmetic mode of averaging gives very These results are summarized in Table.1. The age- similar results for frequencies. The arithmetic mean fre- ing ratio between the travelling and resting clocks is quency hνi of the pulses exchanged during a trip com- p((c + v)/(c − v))v/c, regardless of the Doppler for- posed of diﬀerent stages A and B of homogeneous fre- mula used, relativistic or not. Interestingly using the quencies, is geometric mean, the reciprocal time dilation eﬀect of special relativity completely disappears with the rela- D D hνappi = A ν + B ν (8) tivistic Doppler equation, but not with the conjectural D A D B equation (Table.1). The result calculated here is not the The two phases perceived by the travelling twin are time dilation factor generally reported in the literature vw DA = DB = D, and for the resting twin, as explained [6]. Anecdotally, this result can be written e c2 , an ele- v v previously, D = 1 + D and D = 1 + D. Us- gant combination of velocity v and of hyperbolic rapidity A c B c w = c tanh−1(v/c). ing the three previously used Doppler equations, Eq.(8) gives the results compiled in Table.2.

Table 2: Arithmetic mean frequency averaged over the whole round trip, perceived by the resting (R) and travelling (T ) twins and calculated using diﬀerent Doppler formulas.

Point of view Classical Relativistic Conjectural

νapp 1 1 Traveller RT 1 v2 ν q v2 1 − c2 1 − c2 app ν q 2 2 Inert TR 1 1 − v 1 − v ν c2 c2 hνappi 1 1 1 Ratio RT hνappi v2 v2 v2 TR 1 − c2 1 − c2 1 − c2

Once again, the three Doppler formulas give the same 3 Shortcut veriﬁcations with en- result, close to the previous one. ergy

The objective of this section is twofold: (i) verify the previous result through a diﬀerent approach and (ii) gener- alize the correspondence between waves and time distor- tions to all other situations. Result: A round-trip traveller aged

v s v ! c 3.1 The energetic connection of time and 1 1 + c 2 ∼ v wavelengths 1 − v 1 − c2 c . Following the Planck/Einstein relationship E = hν, fre- times slower than his resting counterpart who stayed quency is energy. After showing identical equations 1 v2 at home. This result is greater by about 2 c2 than the for energy and frequencies, Einstein concluded that fre- q v2 quency and energy vary with the same law with the state currently admitted ageing ratio of 1/ 1 − c2 . of motion of the observer [4]. This correspondence is in fact natural with respect to the quantum of time. In-

3 deed, in the principle of uncertainty ∆E∆t ≥ h/4π, the to an other frame at speed v, has to consume a kinetic energetic component can be expressed as ∆E = h∆ν, energy such that which gives ∆t∆ν ≥ 1/4π, showing that time and frequencies are mutually constrained. This relationship al- νmov E − ∆E mc2 − 1 mv2 1 v2 = jump = 2 = 1 − (10) lows to simply predict Doppler effects if assuming mass ν E mc2 2 c2 conservation. The time dilation of special relativity is recovered: λmov 1 1 = 2 ≈ r (11) 3.2 Mass conservation λ 1 v v2 1 − 2 1 − According to everyday experience, the mass of clocks 2 c c2 (travelling inside rockets), will be supposed unchanged upon arrival. This assumption avoids an internal con- 3.5 Uniform acceleration tradiction in special relativity pointed by de Broglie: A Suppose that two objects A and B of identical masses change of frame predicts a change of energy, from E0 = q m, are submitted to a uniform acceleration field γ shap- 2 mov 2 v2 m0c to E = m0c / 1 − c2 and then to a change ing the whole space and that B is shifted in this field q at a distance d from A. The potential energy of B has of frequency νmov = Emov/h = m c2/h 1 − v2 = 0 c2 decreased compared to that of A q ν / 1 − v2 . This result is inverse to that expected 0 c2 νB EB EA − ∆E from a time dilation perspective. To plug this breach, = = (12) νA EA EA de Broglie developed a complex theory called ”harmony with of the phases” with phase speeds higher than c [7]. As 1 2 1 2 1 2 2 these phases have not been evidenced yet and for simplic- ∆E = mvB − mvA = m(vB − vA) (13) ity, mass effects will be prudently considered negligible 2 2 2 here. This value is given by the Newtonian rules of uniform acceleration: vB = vA + γt (14) 3.3 The round trip (rt) and 1 2 The total energy of the inert twin is its resting energy d = vAt + γt (15) 2 2 E = mc . During his round trip, his brother first jumps By eliminating the variable t between these two equa- in a new frame moving at speed v relative to the starting tions, one obtains one, and then jumps back to his initial frame. 2 2 vB − vA = 2γd (16) νrt Ert E − ∆E = = jumps (9a) As it is impossible to get out of the space-wide acceler- ν E E ation field, the farthest object A will be arbitrarily con- with sidered as the reference one and Eq(12) finally gives νB mc2 − mγd γd 1 2 1 2 2 = = 1 − (17) ∆Ejumps = mv + m(−v) = mv (9b) νA mc2 c2 2 2 This formula is sometimes adapted to the gravita- which gives tional redshift by considering γ as a gravitational acceleration γ = g = GM/r2 where G is the gravitational νrt mc2 − mv2 v2 = = 1 − (9c) constant, r is the radial distance between the source and ν mc2 c2 the center of a graviting body of mass M. But this treatment ignores that a gravitational free fall is not uniform so that but is an accelerated acceleration, as detailed later. λrt 1 1 + v v/2c = ∼ c (9d) λ v2 1 − v 1 − c 3.6 Gravitational redshift c2 3.6.1 Loss of gravitational potential The two approaches (Doppler and energy) give the same results. An object trapped in a gravitational well should consume energy to escape it, because of its loss of potential gravitational energy (GMm/r), compared to an identical 3.4 Uniform motion mass located very far from the well, The total energy of an inert clock is its resting energy νgr E − ∆E 2 = gr (18a) E = mc . Starting from this frame, a clock jumping ν E

4 3.7 The cosmological redshift νgr mc2 − mGM/r GM = = 1 − (18b) ν mc2 rc2 Even before the publication of Hubble [8], Lemaˆıtrehad Using series expansions near zero, shown that wavelengths should follow expansion [9]. For an interval of universe λgr GM 1 ≈ 1 + ≈ (18c) λ rc2 r 2GM 1 − 2 2 2 2 rc2 ds = dt − a(t) dσ (24) known as the redshift of Einstein. where dσ Is the element length of a space of radius equal to 1, the equation of a light beam is 3.6.2 Accelerated gravitational acceleration

Gravity (g) is not an uniform acceleration but is inversely Z t2 dt σ − σ = (25) proportional to the square of the distance r from the cen- 2 1 a ter of the graviting mass M (g = GM/r2). Hence, a free t1 fall in vacuum towards a massive body is an ”accelerated where σ and σ are the coordinates of a source and acceleration”. The distance x(t) crossed by an inert body 1 2 an observer. A beam emitted later at t1+δt1 and arriving jumping at t0 = 0 from an altitude a (va = 0), follows at t2 + δt2 undergoes a shift such that dx t = GM (19) dt (a − x)2 δt δt 2 − 1 = 0 (26) with the initial condition x(0) = 0. Based on this equa- a2 a1 tion, the distance between the jumper and the center of gravity r = a − x, decreases with time according to giving δt2 a2 r 3 z = − 1 = − 1 (27) r(t) = 3 a3 − GMt2 (20) δt1 a1 2 and when r reaches the altitude b (< a), the speed is where δt1 and δt2 can be considered as the periods at vb = γbtb + va where tb is obtained with Eq.(20) and γb emission and reception respectively [9]. If a procession is deduced from Eq.(19), of walkers regularly spaced crosses a stretching rubber band, on arrival their spacing will obviously be stretched r 1 2 in the same ratio as the rubber band. The same reasoning v = GM(a3 − b3) (21) b b2 3 holds for a series of wave crests. In his article, Lemaˆıtre which implies a drop of potential energy of the same called this effect a Doppler effect [9]. This term is ac- amount, generating a redshift of a source B (at altitude ceptable if broadly defining the Doppler effect as a wave b) perceived by a receiver A (at altitude a), of distortion, but this is not the classical Doppler effect re- lated to the speed of the source. The ratio between the νB GM a3 − b3 reception and emission wavelengths simply follows the = 1 − (22) νA 3c2 b4 increase of the distance D between the source and the This gravitational redshift between two objects in the receiver, which took place during the travel of light: same well is not an absolute value but depends on the relative altitude of the source and the receiver. If both ∆tapp T app D = = reception (28) are at the same distance from the massive object (a = b), ∆t T Demission no shift is detected. The usual treatments of the gravi- 2 tational redshift using Eq.(17) with γ = GM/r , yields This redshift is exclusively a phenomenon of wave dis- different results. The comparison with the present treat- tortion, holding even if the sources and the receiver be- ment gives the following value to r: long to the same inertial frame. The association between the redshift and the duration [1], is the ultimate proof of r a − b r = b2 3 (23) the connection between time and wavelength distortion. a3 − b3 The connection with energy is less obvious than for the which shows that the two approaches are equivalent previous examples of time distortion, but exists when only for r = a = b, that is to say without jump. The vir- considering the relationship between space and energy. tual radius r is in fact lower than b, because the free fall The uncertainty principle links space and momentum in is an accelerated acceleration. This overlooked property ∆x∆p ≥ h/4π, where p = E/c. Hence, ∆E ≥ hc/4π∆x. forbids coupling Eq.(17) to γ = GM/r2. Stretching x implies a decrease of E to maintain this fundamental relationship.

5 4 Conclusion rameters. Proceedings of the XIX International Sym- posium on Lepton and Photon Interactions at High Strangely, the popular twin paradox is sometimes consid- Energies. Stanford, California, 1999. ered as a verification of the time dilation of special relativity, whereas special relativity deals only with uniform [2] Michel, D. 2015. Slowing time by stretching the waves motion. Doppler analyses predict different time distor- in special relativity. hal-01097004v6. tion effects depending on whether the moving source only passes near the receiver or starts from the receiver frame [3] Darwin, C.G. 1957. The clock paradox in relativity. and returns to its point of origin. The symmetrical time Nature 180, 976-977. dilation of special relativity based on the size of time units, is radically different from the non-symmetrical [4] Einstein, A. 1905. Zur Elektrodynamik bewegter time dilation of the twin paradox in which the number Körper (On the electrodynamics of moving bodies) of time units differs between the observers. The twin Annal. Phys. 17, 891-921. thought experiment mixes (i) a traditional Doppler effect stretching and then shortening the standard wavelength [5] Einstein, A. 1918. Dialog über Einwändegegen die and (ii) a shift in the perception of the about-turn. The Relativitätstheorie.Die Naturwissenschaften. new solution of the twin paradox proposed here can be obtained in different ways. This alternative solution dif- [6] Lasky, R.C. 2012. Time and the twin paradox. Sci. fers from the classical result that is built on the Lorentz Am. 16, 21-23. dilation factor. In practice, both formulas give similar but possibly testable results. This new solution is based [7] de Broglie,L. Recherches sur la théoriedes quanta. on the conception of time as a frequency depending on Thèsede physique. Paris, 1924. the energetic status of the system and its validity is sup- ported by the general energetic approach to wavelength [8] Hubble, E. 1929. A relation between distance and ra- and time distortion effects in all fields of physics. dial velocity among extra-galactic nebulae. Pnas 15, 168-173.

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