Twin Paradox” Experiment, Consistent with the Time-Energy Relationship Denis Michel

Prediction of a new result for the ”twin paradox” experiment, consistent with the time-energy relationship Denis Michel

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Denis Michel. Prediction of a new result for the ”twin paradox” experiment, consistent with the time-energy relationship. 2015. hal-01182589v2

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Denis Michel

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

Abstract symmetry of special relativity is broken by the moving twin, even he flies at constant speed. Einstein himself A new result is proposed for the time dilation effect was troubled by the consequences of his own work. He of the round trip, known as the twin paradox thought first explained the experience of half-turn of a clock us- experiment. This differential ageing effect has strangely ing the time dilation of special relativity [6]. Later, been treated in general relativity in absence of gravity he attributed the asymmetry between the clocks to a and in special relativity in absence of uniform motion, pseudo-gravitational effect of the about turn [7]. How- which misleadingly inspired the idea of paradox. The ever, atomic clocks have been shown insensitive to ac- alternative Doppler treatment proposed here reveals a celeration [5]. In fact as suggested below, the result of new solution unrelated to the Lorentz dilation factor. this experiment does not require the relativity theory This solution is then shown consistent with the general and can be found using the classical Doppler equation. time-energy relationship which applies to all cases of The wrong treatment of the twin experiment would be time distortion, including special relativity, gravity and to use special relativity tools (which hold exclusively for cosmological expansion. inertial frames) and to arbitrarily decide who travels and who remains at rest. This can be readily shown: Twin brothers are separated; one of them remains inert while the other crosses a distance dl before joining his brother. 1 Introduction As they 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 t′ who travels, crease, or equivalently frequency decrease. This phe- ′ nomenon is particularly obvious for distant supernovae ds2 = (cdt)2 = (cdt )2 − dl2 (1a) app whose large redshift (for example λ /λ = 1.5), pre- with v = dl/dt′, one obtains the traditional time dila- cisely coincides with an increased duration of brightness ′ v2 (1.5-fold longer in the example) [1]. In special relativity tion result dt /dt = 1/ 1 − c2 , and conversely if it is (uniform motions), a time interval corresponds to a given the other twin (of properq time t) who travels, ′ number of periods (say n) so that ∆t > ∆t is equivalent 2 2 2 ′ 2 ′ ′ ds = (cdt) − dl = (cdt ) (1b) to nT > nT and to ν < ν. In other words, wave distortion exactly compensates spacetime distortion in such ′ v2 with v = dl/dt, dt/dt = 1/ 1 − 2 . Hence, it is im- a way that (i) the speed of light and (ii) the number of c periods, are both maintained identical for all observers possible to break the symmetry.q No physical experiment [2]. In addition in the twin paradox, following the idea of can measure the absolute velocity of a frame with a uni- Darwin [3], there is a difference in the number of periods. form motion in which it is conducted, so that one can This phenomenon generates a new differential ageing ef- not assign the speed v specifically to one twin. This fect irrespective of the Doppler formula used, showing problem is at the origin of the term paradox, but now that it is not specifically a matter of special relativity. most authors point that in fact, the moving twin does Finally, the time-energy relationship is shown to predict not remain in a single inertial frame since his motion is and unify all the types of time distortion. not uniform, and certain authors recourse to general relativity to treat this situation. Many resolutions of this actively debated experiment have already been described 2 The twin experiment in the context of either special or general relativity. The most frequently reported result is that upon arrival, the Contrary to special relativity in which all the frames clock of the travelling twin delays relative to that of the v2 are uniformly moving, in the famous twin paradox, the resting twin in the ratio 1 − c2 [3, 4, 5], for example q 1 0.8 for a travel at a speed of 0.6 c [4]. A different re- tinguishing between the relativistic and classical Doppler sult is proposed here. To simplify the treatment, let us effects. The number of wave crests received by the inert consider a collinear round trip (starting from a spatial twin is lower than that received by the travelling twin, station to eliminate a role for gravity). When located which means that the symmetry is broken contrary to at a distance D from his sedentary brother, the trav- special relativity in which the number of pulses is the elling twin makes an about-turn at constant speed and same for all observers. infinite acceleration (like a frontal elastic collision). The In the numerous debates on the twin paradox, a fre- twins continuously exchange light pulses with the same quent argument against the followers of general relativ- fundamental wavelength at the origin. Viewed by the ity is that the phase of acceleration or pseudo-gravity travelling twin, things are very simple. He perceives in- can be as short as desired compared to the duration of stantaneously his about-turn and sees a dilated Doppler the trip. We see in the mechanism described here that effect from his resting brother before his turn-around and this phase can indeed be reduced to zero in an instan- a shrinked Doppler effect during the trip back. Viewed taneous about-turn like an elastic collision, but the time by the resting twin, things seem simple in appearance but period of asymmetry remains equal to the fraction v/c are less simple in reality because as explained in [3], he of the trip. For his demonstration, Darwin used the rel- cannot perceive the change of Doppler effect during the ativistic Doppler equation and the arithmetic mean [3]. turn because light emitted at this point takes a time D/c The arithmetic and geometric modes of averaging will to reach him, so that when he perceives it, the travelling be tested here with different Doppler equations. The re- twin has already crossed d = vD/c towards him. The sult will be shown independent on the Doppler equation switch between the blue and red shifts does not occur at used. Since the geometric mean is more appropriate for v D but at D + d, or D 1+ c . This mere fact explains averaging simultaneously wavelengths and frequencies, it why the number of pulses received by the twins is not will be used first. symmetrical. Hence, the twin paradox can not help dis-

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 the round trip.

Point of view Classical Relativistic Conjectural

app λRT v2 v2 Traveller 1 − 2 1 1/ 1 − 2 λ c c v v v app qv c v c vq c λTR 1+ c v2 1+ c 1+ c v2 Inert v 1 − c2 v v / 1 − c2 λ s1 − c ! s1 − c ! s1 − c ! q v v qv app v c v c v c λTR 1+ c 1+ c 1+ c Ratio app v v v λRT s1 − c ! s1 − c ! s1 − c !

2.1 Wavelength geometric averaging 2.1.1 Home clock perceived by the traveller Using the classical Doppler formula,

1 app 1 2 2 λ v v 2 v RT = 1+ 1 − = 1 − (2) λ c c c2 app λRT is the wavelength of the resting source viewed by app with the relativistic Doppler formula, the traveller and λTR is the wavelength of the travelling source viewed by the resting twin. The asymme- 1 2 try between the twins will be evaluated using either the app v v λRT 1+ c 1 − c classical, the relativistic, or the conjectural [2] Doppler = v v = 1 (3) λ s1 − c s1+ c ! formulas, on the basis that the Doppler eﬀects perceived v by the resting twin is dilated during the 1+ c th of the and using the conjectural Doppler formula, journey and is shrinked during the 1 − v th of the jour- c − 1 app − 1 2 2 ney. Wavelengths can be averaged geometrically over the λ v v 2 v RT = 1+ 1 − = 1 − (4) whole trip as follows. λ c c c2 2 2.1.2 Traveller’s clock perceived by the resting special relativity completely disappears with the rela- brother tivistic Doppler equation, but not with the conjectural equation (Table.1). The result calculated here is not the Using the classical Doppler formula, time dilation factor generally reported in the literature 2 1 evw/c app v − v 2 [4]. Anecdotally, this result can be written , an λ v (1+ c ) v (1 c ) TR = 1+ 1 − elegant combination of velocity v and of hyperbolic ra- − λ c c pidity w = c tanh 1(v/c). 1 (5) 1+ v v/2c v2 2 = c 1 − 1 − v c2 c 2.2 Arithmetic mean frequency with the relativistic Doppler formula,

1 Apparent temporal paradoxes often disappear if conceiv- v − v 4 app v (1+ c ) v (1 c ) ing time as a frequency depending on the energetic status λTR 1+ c 1 − c = v v of the system. Frequencies provide a very intuitive idea λ 1 − c 1+ c ! (6) of time when applied for example to heart beats, days v v/2c 1+ c and seasons. As stated above, the appropriate tool for = v 1 − c averaging both wavelengths and frequencies is the geometric mean, but the traditional arithmetic mode of av- and using the conjectural Doppler formula, eraging gives very similar results for frequencies. The − 1 arithmetic mean frequency ν of the pulses exchanged app v − v 2 λ v (1+ c ) v (1 c ) TR = 1 − 1+ during a trip composed of diﬀerent stages A and B of λ c c homogeneous frequencies, is − 1 (7) v v/2c 2 2 1+ c v = 1 − app DA DB 1 − v c2 ν = νA + νB (8) c D D These results are summarized in Table.1. The two phases perceived by the travelling twin are The ageing ratio between the travelling and resting DA = DB = D, and for the resting twin, as explained v/c v v clocks is ((c + v)/(c − v)) , regardless of the Doppler above, DA = 1+ c D and DB = 1+ c D. Using the formula used, relativistic or not. Interestingly using the three previously used Doppler equations, Eq.(8) gives the p geometric mean, the reciprocal time dilation eﬀect of 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

app νRT 1 1 Traveller 2 1 v 2 ν 1 − 2 v c 1 − c2 app 2 νTR q v2 v Inert 1 1 − 2 1 − 2 ν c c app q νRT 1 1 1 Ratio 2 2 2 νapp v v v TR 1 − c2 1 − c2 1 − c2

Once again, the three Doppler formulas give the same Result: A round-trip traveller aged result, close to the previous one. v v c 1+ c 1 v ∼ v2 s1 − c ! 1 − 2 c . times slower than his resting counterpart who stayed 2 1 v at home. This result is greater by about 2 c2 than the v2 currently admitted ageing ratio of 1/ 1 − c2 . q 3 with 1 1 ∆E = mv2 + m(−v)2 = mv2 (9b) jumps 2 2 3 Shortcut verifications with en- which gives ergy νrt mc2 − mv2 v2 = 2 =1 − 2 (9c) The objective of this section is twofold: (i) verify the pre- ν mc c vious result through a different approach and (ii) gener- so that alize the correspondence between waves and time distor- λrt 1 1+ v v/2c tions to all other situations. = ∼ c (9d) λ v2 1 − v 1 − c c2 3.1 The energetic connection of time and The two approaches (Doppler and energy) give the same wavelengths result. Following the Planck/Einstein relationship E = hν, frequency is energy. After showing identical equations 3.4 Uniform motion for energy and frequencies, Einstein concluded that fre- The total energy of an inert clock is its resting energy quency and energy vary with the same law with the state E = mc2. Starting from this frame, a clock jumping of motion of the observer [6]. This correspondence is in to an other frame at speed v, has to consume a kinetic fact natural with respect to the quantum of time. In- energy such that deed, in the principle of uncertainty ∆E∆t ≥ h/4π, the energetic component can be expressed as ∆E = h∆ν, which gives ∆t∆ν ≥ 1/4π, showing that time and fre- νmov E − ∆E mc2 − 1 mv2 1 v2 = jump = 2 =1 − (10) quencies are mutually constrained. This relationship al- ν E mc2 2 c2 lows to simply predict Doppler effects if assuming mass conservation. The time dilation of special relativity is recovered: λmov 1 1 3.2 Mass conservation = 2 ≈ (11) λ 1 v v2 1 − 1 − 2 c2 2 According to everyday experience, the mass of clocks r c (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 = 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 mov qmov 2 v2 at a distance d from A. The potential energy of B has of frequency ν = E /h = m0c /h 1 − 2 = c decreased compared to that of A v2 q ν0/ 1 − c2 . This result is inverse to that expected νB EB EA − ∆E fromq a time dilation perspective. To plug this breach, = = (12) de Broglie developed a complex theory called ”harmony νA EA EA of the phases” with phase speeds higher than c [8]. As with these phases have not been evidenced yet and for simplic- 1 2 1 2 1 2 2 ity, mass effects will be prudently considered negligible ∆E = mvB − mvA = m(vB − vA) (13) here. 2 2 2 This value is given by the Newtonian rules of uniform 3.3 The twin experiment acceleration: vB = vA + γt (14) The total energy of the inert twin is its resting energy 2 and E = mc . During his round trip, his brother first jumps 1 in a new frame moving at speed v relative to the starting d = v t + γt2 (15) A 2 one, and then jumps back to his initial frame. The round trip (rt) frequency follows By eliminating the variable t between these two equations, one obtains rt rt ν E E − ∆Ejumps = = (9a) v2 − v2 =2γd (16) ν E E B A

4 As it is impossible to get out of the space-wide acceler- which implies a drop of potential energy of the same ation field, the farthest object A will be arbitrarily con- amount, generating a redshift of a source B (at altitude sidered as the reference one and Eq(12) finally gives b) perceived by a receiver A (at altitude a), of B 2 ν mc − mγd γd B 3 3 = =1 − (17) ν GM a − b A 2 2 =1 − (22) ν mc c νA 3c2 b4 This formula is sometimes adapted to the gravita- This gravitational redshift between two objects in the tional redshift by considering γ as a gravitational accel- 2 same well is not an absolute value but depends on the eration γ = g = GM/r where G is the gravitational relative altitude of the source and the receiver. If both constant, r is the radial distance between the source and are at the same distance from the massive object (a = b), the center of a graviting body of mass M. But this treat- no shift is detected. The usual treatments of the gravi- ment ignores that a gravitational free fall is not uniform tational redshift using Eq.(17) with γ = GM/r2, yields but is an accelerated acceleration, as detailed later. different results. The comparison with the present treatment gives the following value to r: 3.6 Gravitational redshift a − b 3.6.1 Loss of gravitational potential 2 r = b 3 3 3 (23) r a − b An object trapped in a gravitational well should con- which shows that the two approaches are equivalent sume energy to escape it, because of its loss of potential only for r = a = b, that is to say without jump. The vir- gravitational energy (GMm/r), compared to an identical tual radius r is in fact lower than b, because the free fall mass located very far from the well, is an accelerated acceleration. This overlooked property 2 νgr E − ∆E forbids coupling Eq.(17) to γ = GM/r . = gr (18a) ν E νgr mc2 − mGM/r GM = =1 − (18b) 3.7 The cosmological redshift ν mc2 rc2 Using series expansions near zero, Even before the publication of Hubble [9], Lemaˆıtre had λgr GM 1 shown that wavelengths should follow expansion [10]. For ≈ 1+ ≈ (18c) an interval of universe λ rc2 2GM 1 − 2 r rc ds2 = dt2 − a(t)2dσ2 (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 t2 Gravity (g) is not an uniform acceleration field but is in- dt σ2 − σ1 = (25) versely proportional to the square of the distance r from 1 a Zt the center of the graviting mass M (g = GM/r2). Hence, where σ1 and σ2 are the coordinates of a source and a free fall in vacuum towards a massive body is an ”ac- an observer. A beam emitted later at t1+δt1 and arriving celerated acceleration”. The distance x(t) crossed by an at t2 + δt2 undergoes a shift such that inert body jumping at t0 = 0 from an altitude a (va = 0), follows δt δt 2 − 1 = 0 (26) a a dx t 2 1 = GM (19) dt (a − x)2 giving δt2 a2 with the initial condition x(0) = 0. Based on this equa- z = − 1= − 1 (27) δt a tion, the distance between the jumper and the center of 1 1 gravity r = a − x, decreases with time according to where δt1 and δt2 can be considered as the periods at emission and reception respectively [10]. If a procession of walkers regularly spaced crosses a stretching rubber 3 3 3 2 r(t)= a − GMt (20) band, on arrival their spacing will obviously be stretched r 2 and when r reaches the altitude b (< a), the speed is in the same ratio as the rubber band. The same reasoning holds for a series of wave crests. In his article, Lemaˆıtre vb = γbtb + va where tb is obtained with Eq.(20) and γb is deduced from Eq.(19), called this effect a Doppler effect [10]. This term is ac- ceptable if broadly defining the Doppler effect as a wave distortion, but this is not the classical Doppler effect re- 1 2 3 3 vb = 2 GM(a − b ) (21) lated to the speed of the source. The ratio between the b r3

5 reception and emission wavelengths simply follows the distortion effects in all fields of physics. This theoreti- increase of the distance D between the source and the cal prediction is submitted to future precise experimental receiver, which took place during the travel of light: tests, in particular to distinguish the difference of v2/2c2 with the classically admitted result. ∆tapp T app D = = reception (28) ∆t T Demission This redshift is exclusively a phenomenon of wave dis- References tortion, holding even if the sources and the receiver be- long to the same inertial frame. The association between [1] Perlmutter, S. Supernovae, dark energy, and the ac- the redshift and the duration [1], is the ultimate proof of celerating universe: The status of the cosmological pa- the connection between time and wavelength distortion. rameters. Proceedings of the XIX International Sym- The connection with energy is less obvious than for the posium on Lepton and Photon Interactions at High previous examples of time distortion, but exists when Energies. Stanford, California, 1999. considering the relationship between space and energy. [2] Michel, D. 2015. Slowing time by stretching The uncertainty principle links space and momentum in the waves in special relativity. https://hal.archives- ∆x∆p ≥ h/4π, where p = E/c. Hence, ∆E ≥ hc/4π∆x. ouvertes.fr/hal-01097004v6/document Stretching x implies a decrease of E to maintain this fundamental relationship. [3] Darwin, C.G. 1957. The clock paradox in relativity. Nature 180, 976-977.

4 Conclusion [4] Lasky, R.C. 2012. Time and the twin paradox. Sci. Am. 16, 21-23. Strangely, the popular twin paradox is sometimes considered as a verification of the time dilation of special rela- [5] Lämmerzhal, C. 2005. Special relativity and Lorentz tivity, whereas special relativity deals only with uniform invariance. Ann. Phys. 14, 71-102. motion. Doppler analyses predict different time distortion effects depending on whether the moving source only [6] Einstein, A. 1905. Zur Elektrodynamik bewegter passes near the receiver or starts from the receiver frame Körper (On the electrodynamics of moving bodies) and returns to its point of origin. The symmetrical time Annal. Phys. 17, 891-921. dilation of special relativity based on the size of time units, is radically different from the non-symmetrical [7] Einstein, A. 1918. Dialog über Einwände gegen die time dilation of the twin paradox based on the number Relativitätstheorie. Die Naturwissenschaften. of time units. The twin thought experiment mixes (i) a [8] de Broglie,L. Recherches sur la théorie des quanta. traditional Doppler effect stretching and then shortening Thèse de physique. Paris, 1924. the standard wavelength and (ii) a shift in the percep- tion of the about-turn. The new solution of the twin [9] Hubble, E. 1929. A relation between distance and ra- experiment obtained here in different ways, differs from dial velocity among extra-galactic nebulae. Pnas 15, the classical result that is built on the Lorentz dilation 168-173. factor. In practice, both formulas give similar but pos- sibly testable results. This new solution is based on the [10] Lemaˆıtre G. 1927. Un Univers homogène de masse conception of time as a frequency depending on the ener- constante et de rayon croissant rendant compte de la getic status of the system and its validity is supported by vitesse radiale des nébuleuses extra-galactiques. An- the general energetic approach to wavelength and time nal. Soc. Sci. Bruxelles. A47, 49-59.