The physics of the longitudinal light clock

Transcription

1 he physis of the longitdinal light lok Giovanni Zanella Stdioso Senior dello Stdim Patavinm Università di Padova, Italy bstrat he standard analysis of the behavior of the longitdinal light lok, in inertial motion, is not onvining in order to respet the postlates of the Speial Relativity. Indeed, the time neessary to the light to travel from a mirror to the other in the moving lok is the same for a omoving observer, while it appears asymmetrial to a stationary observer. On the ontrary, sh asymmetry an be restored if the Doppler effet of the light emitted from a moving sore is extended also to spae and time. s a onseqene, the known orentz s ontration of the bodies in inertial motion is pt in disssion. 1. Introdtion he standard analysis of the behavior of a longitdinal light lok (C) in niform translatory motion (Fig.1), sing the methods of the theory of the Speial Relativity (SR), shows an asymmetry whih is not in aord with the two postlatees of SR. We reall here these postlates as expressed by. Einstein [1]: 1. he laws by whih the states of physial systems ndergo hange are not affeted, whether these hanges of state be referred to the one or the other of two systems of o-ordinates in niform translatory motion.. ny ray of light moves in the stationary system of o-ordinates with the determined veloity, whether the ray be emitted by a stationary or by a moving body. ording with the priniple of relativity, the first postlate tells s that an observer within a ab in niform motion of translation with respet to a stationary referene, annot with experiments internal to the ab to disover if the ab is moving, or not. On the other hand, an observer at rest in the 1

2 stationary referene annot disern the motion of the ab only looking to the physial phenomena whih appear inside the ab itself. he seond priniple tells s, differently from the Galileo s view, that the motion of a sore of light annot be imparted to the emitted light from the sore itself. s onerns the time, it an be intended as the path of a ray of light along a gradated rigid axis. Indeed, being and two points of the spae where are plaed two loks, Einstein wrote [1]: we establish by definition that the time reqired by light to travel from to eqals the time it reqires to travel from to. et a ray of light start at the time t from towards, let at the time t be refleted at in the diretion of, and arrive again at at time time t. In agreement with experiene we frther assme the qantity t t (1) to be a niversal onstant: the veloity of light in empty spae. In aordane with definition the two loks synhronize if t t t t. () Obviosly, with this proedre it is the rate of the two loks that is tned and not the start point of the time marked by them. herefore, t t of Eq.(1) expresses the time spent by the light ray to travel over the path --, so ( t t )/ expresses indifferently the time spent by the light ray over the path or. hs, if onventionally we pt 1, the path of a ray light represents too the time neessary to the light to ross the path itself, in any inertial referene. Starting from these premises, we analyze the behavior of the moving C showing the neessity of an extension of the Doppler effet also to spae and time involved by the moving light sore. In the following analysis the observer is spposed biqitos in the spae asribed to his referene system, as also Einstein intended. esides, we intend the C to have a mass so high to neglet reoil effets de to the refletions of the ray of light on the two mirrors.

3 . Galilean view Fig. shows the shema of the C in niform translatory motion (or inertial motion) with veloity with respet to a stationary referene, in a Galilean senario. he Galilean senario intends time and spae to be absolte and besides it onsiders the omposition of the veloity of the light with that of the C. herefore, the time neessary to the ray of light to travel from end to end, as viewed from the stationary referene, will be + + +, that is (3) where is the proper distane of the mirrors, or the proper length of the rod of spport. t the same manner, the time neessary to the ray of light to travel from to, will be, that is (4) herefore, the first priniple of SR is respeted in the Galilean senario of C, bease looking to the internal behavior of C from the stationary it is not possible to disern the motion of the lok itself. Obviosly, the same reslt appears when the ray of light is replaed by a small sphere of negligible mass whih is refleted elastially between the mirrors. 3. he standard analysis of C Introding the seond priniple of SR, that is the invariane of the veloity of the light in the empty spae, the time reqired to a light ray to ross the path (Fig.3), spposing << (orentz s fator γ 1), wold be 3

4 +, that is, (5) while for the path, that is +. (6) We an note that also Einstein arrived to the same reslt of Eq.(5) and Eq.(6) by a thoght experiment [1]. He, sed rays of light and plaed two synhronized loks (with the same start of the time) at the ends and of a rigid rod. So, imparted to the rod a niform motion of parallel translation with veloity, along the diretion of its axis, he wrote [1]: et a ray of light depart from at the time t, let it be refleted at at the time t, and reah again at the time t. aking in onsideration the priniple of the onstany of the veloity of light we find t t and t t (7) + where denotes the length of the moving rod, measred in the stationary system. Observers moving with the moving rod wold ths find that the two loks were not synhronos, while observers in the stationary system wold delare the loks to be synhronos. Here the mistake of Einstein appears evident, bease a omoving observer with the rod annot have means to detet its inertial motion. Sbseqently he repaired the mistake pointing ot that the asymmetry was deteted only by the stationary observer looking to the moving rod []. We known, instead, that this asymmetry is still against the first postlate of SR, as well as the omposition of the veloity of the light with the veloity is against the seond postlate of SR. 4

5 herefore, if the standard analysis of C doesnt ondt to a reslt physially aeptable it appears evident that frther nderlying physis mst be explored. s we will see, a frther rak of the Einstein s view will appear when the known orentz s ontration of the bodies in inertial motion goes in ontrast with the dilation of the time [3]. 4. Physis of C Sppose that S and S are two Cartesian referene systems having parallel axes, where the x -axis is in niform translation with veloity << (γ 1) along the positive x-axis (Fig.4). he time started when the origin O of S and the origin O of S were in oinidene (t 0). Fig.4 shows the snapshot, at the time t, of the two dimension wave-front of light generated in the time t0, and in the empty spae, from a isotropi sore loated in the oinidene point of the origins of S and S. Now, the wave-front of Fig.4 is neessarily a onentri sphere with O, if viewed from a omoving observer with S. his eventality is possible only if the Doppler effet involved by the motion of O is extended also to the spae and onseqently to the time. In pratie, the spae beomes ompressed in front to O, in the diretion of, and dilated in the opposite diretion, as viewed from the stationary referene [3]. herefore, with respet to O, as viewed from S, two paths are assoiated to the wave-front whih propagates along the x-axis (solid arrows of Fig.4): the path X f orresponding to the forward path along the diretion of the motion of S and the path X b asribed to the opposite diretion. So, the path X f reslts a ontration of Xt, that is (Fig.4) X t X X X f 1, (8) and the path X b reslts a dilation of Xt, that is (Fig.4) + X + t X + X X b 1, (9) Hene, looking to Eq,(5) and Eq.(6), we mst ontrat with the fator ( ) /, when the light ray travels forward from to and dilate the 5

6 spae with the fator ( + ) /, when the light travels bak from to. So, the times and beome and + +. (10) In pratie, we have obtained, independently on the veloity, and also the onstany of the veloity of the light in either diretions of motion (Fig.5). On the other hand, introding the known orentz s dilation on and [1], we have / 1 / / 1 / / 1 /. (11) Eq.(11) ompels s to dilate in / 1 /, in aord with the notion of the time intended as the path of a light ray (Fig.5). It is remarkable that in this relativisti view a same sale fator (the orentz s fator 1/ 1 / ) pertains the whole moving system so that the observer, omoving with the lok, has not means to disern its inertial motion. In lak of the Doppler effet on spae and time, we wold find as overall time for the light ray, to over a forward and bak path, the time + +, (1) so 1, bt for SR shortened length of C, that is 7. Conlsions, and then we wold have a 1 1. (13) he standard analysis of the behavior of the longitdinal light lok, in niform translatory motion with respet to a stationary referene, reveals an asymmetry of the time against the first priniple of Speial Relativity [1]. Indeed, the time neessary to a light ray to travel from a mirror to the other 6

7 wold reslt dilated in the forward path, along the diretion of the motion of the lok, and shortened in the bak path. s a onseqene, it wold be possible to disern the inertial motion of the lok from a stationary referene, looking to the internal behavior of the lok itself. he paper shows instead that sh asymmetry does not exist, bease it an be restored if the Doppler effet is extended also to the spae and to the time involved by the moving light sore [3]. t least, introding the known relativisti time dilation, by the orentz s fator, also the length of the longitdinal light lok mst dilate at the same manner, against the known orentz s ontration. Referenes 1.. Einstein, Zr eletrodynamik bewegter orper, nn. Phys. 17 (1905) Einstein, Slla teoria speiale e generale della Relatività, Zanihelli Editore, ologna (191). 3. G. Zanella, Doppler effet of time and spae, Prespaetime J. 3, 9 (01) and Figre aptions Fig.1. Shemati representation of a longitdinal light lok (C). Fig.. Galilean view of the behavior of C in niform translatory motion with veloity with respet to a stationary referene (see text). Fig.3. Shemati representation of C of Fig.1 in the standard senario with << (see text). Fig.4.wo-dimensional representation of the wave-front of light emitted isotropially from a sore in niform translatory motion the empty spae as viewed from a stationary referene S. he sore, loated in the origin of the system S, moves with veloity 0.5 ( orentz s fatorγ 1) with respet to system S (see text). Fig.5. Shemati representation of C of Fig.1 in a relativisti senario, where γ represents the orentz s fator (see text). 7

8 Refleting mirrors Rigid rod Rays of light Fig. 1 t + - t Stationary referene Fig. 8

9 t t Fig. 3 Stationary referene Wave-front of light in the time t S S t X b O O t X f X x x Stationary referene Flash of light in the time t 0 Fig. 4 9

10 t γ γ t Fig. 5 Stationary referene 10