The Mechanics of Adding Velocities 2011 Robert D. Tieman

Transcription

1 The Mehais of Addig Veloities 011 Robert D. Tiema We must ow aalyze the qualities assoiated with addig eloities with respet to our urret uderstadig of the mehais of motio. We begi by aalyzig that whih is gie by most relatiisti tets as proof of the relatiisti additio of eloities. I. Eperimetal Additio of Veloities I the year 1818, Augusti Fresel deried a equatio detailig the amout by whih light would be dragged by the ether, where the ether was treated as beig partially dragged by a refratie medium. 1 I-1 1 This equatio was erified withi eperimetal limits by Hippolyte Fizeau i 1851 ad Martius Hoek i 1868 utilizig a iterferometry based eperimet reliat upo how light propagates through a refratie medium. The relatiisti ofirmatio of the ompoudig of eloities as ofirmed by suh eperimets is otroersial. t y y I- z z t t The preedig equatios are gie by Eistei as the Loretz trasformatio of oordiates betwee differig frames of referee separated by a relatie eloity. I aordae with the rest frame, we utilize the ierse Loretz trasformatios. 1

2 t y y I-3 z z t t The resultig eloity with respet to a rest frame is thus gie as follows. t u t t t u t t u, u t 1 t u I-4 u u 1 The ierse relatio is gie as follows. u I-5 u u 1 Now we merely allow a approimatio to take plae with respet to this equatio. 1 u u u 1, u 1 u 1 The Biomial Theorem epressed as a Taylor series epasio is defied as follows.

3 1 1 a b a a b a b... b Therefore, we a epad the relatiisti eloity equatios as follows. u 1 u Igorig seod order effets with respet to the eloity, we a further approimate this relatioship. u 1 I-6 u 1 This is the relatio as deried by Ma Vo Laue i The equatio gie by Fresel for the ether drag oeffiiet is of ourse equal to this relatio. The relatiisti equatio for eloity is based upo eletromageti waes traersig a uiform spatial medium oid of ay eteral ifluee to ilude eletri ad mageti fields of fore. It is iterestig to see how the Loretz trasformatios predit suh a odd effet for light propagatig with respet to a refratie medium. How a a mehaisti relatiisti epressio for ompoudig eloities predit the etraimet of light by a refratie medium? It beomes readily apparet that there is ot a adequate aswer for this questio, oly a mathematial solutio. Let us ow rewid ad larify the deriatio as gie by Ma o Laue. We begi with the approimated relatiisti eloity additio. u I-7 u u 1 The substitutio of u with allows the Fresel equatio to be deried. u u u 1, u 3

4 u 1 1 u 1 What is importat to ote is that the eloity of light i the deomiator is retaied durig this substitutio, yet the equatio is iteded to desribe a eet withi a refratie medium where the eloity of light is altered. I order to address this mathematial uriosity, we ow tur our attetio to the aepted appliatio of the relatiisti additio of eloities. z z u y y K K Figure 1 The preedig figure illustrates a eet ourrig withi a auum free of eteral ifluees as gie by the speial theory of relatiity. As the preedig figure illustrates, the eloity of a eet with respet to the iertial frame is u as obsered by a iertial obserer K. Therefore, the followig equatios of speial relatiity theory apply. u u u 1 u u u 1 4

5 Based upo this fudametal mehaisti defiitio, it beomes apparet that the etraimet of light by a refratie medium aot be predited by suh a equatio. It is also a fat that wheeer the eloity u or u is asribed to a eletromageti wae, the the equatios will always redue to the eloity due to the origial desig of the Loretz trasformatios based upo the seod postulate. u 1 u 1 We a further iestigate the ature of the relatiisti additio of eloities by realizig the followig seario. z u y K Figure Suppose the preious figure illustrates a eletromageti eet that takes plae withi a uiform refratie medium that is at rest, where the ide of refratio is greater tha uity (>1). The resultig eloity of the moohromati eletromageti wae is gie as follows. 5

6 u Now we allow this moohromati eletromageti wae to be emitted from a iertial system, while the uiform refratie medium still remais at rest. z z k u y y K K Figure 3 Due to the seod postulate, the rest frame K must obsere that this eloity remais ostat withi this uiform refratie medium regardless of the eloity of the soure of this emissio. We must ote that this relatioship would oly be true i a approimate sese due to the relatiisti Doppler effet (the ide of refratio would hage slightly due to the hage i waelegth ) alterig the resultig waelegth of the emitted wae, whih would slightly alter the ide of refratio ad the eloity of the eletromageti wae for the partiular waelegth ad is a osideratio that must be take ito aout whe aalyzig the Fizeau eperimet i light of the relatiisti additio of eloities. 6

7 z u? y K Figure 4 As the preedig figure suggests, if we ow allow the refratie medium to obtai a etor eloity opposite ad equal to the preious seario, the what would be the resultig eloity of the emitted eletromageti wae? From a mehaisti stadpoit, appliatio of the first postulate of the speial theory of relatiity would demad that the results be equal to the preious seario. There is urretly o mehaial desriptio of the simple additio of eloities that would justify the etraimet of the eletromageti wae withi the refratie medium. I aordae with the first postulate, we would hae to madate that the eloity of the eletromageti wae must remai just as it was i the preious seario, with the oly differee beig the referee that we attah the relatie eloity to. Relatiisti reiproity demads that both searios produe the same results. Therefore, the followig relatio must hold true. u 1 7

8 Clearly, this relatioship oly holds true for eletromageti waes propagatig i a auum. This result idiates that a ujustified assumptio has bee made withi the alulatios. I order to determie this assumptio, we merely re-ealuate this relatioship statig oly the kow ariables (sigle ariable substitutios) sie we a diretly erify their legitimay

9 I-8 Therefore, the followig relatio must hold true i aordae with the first postulate. I-9 u 1 I order for us to apply the relatiisti additio of eloities to this seario, we must be uiform i our substitutios of the eloity of light. The substitutio made by Ma o Laue i order to approimately derie the Fresel relatio was i error due to improper substitutio of the ariables. Whether we hoose for the soure of light or the refratie medium to possess the motio, the Doppler effet must be take ito osideratio as well, whih is further proof of why the relatiisti additio of eloities aot be applied to suh a seario. The problem with applyig Eistei s eloity additio equatio to suh a seario was reogized by Curt Reshaw ad detailed withi his paper The Eperimet of Fizeau as a Test of Relatie Simultaeity. The mai fat is that the relatiisti additio of eloities, or ay equatio of basi mehais detailig the additio of eloities, is ot eough to desribe ay suh eperimet. The appearae of etraimet is merely a illusio reated by the Doppler effet. 9

10 II. Proper Additio of Veloities I aordae with iertial relatiity, the followig equatio defies the relatiisti additio of eloities. u u u 1 We ow retur to the disussio as gie preiously withi this artile with regards to the proper utilizatio of this equatio. z z u y y K K Figure 5 The preedig figure illustrates a eet ourrig withi a auum free of eteral ifluees as gie by the speial theory of relatiity. As gie preiously, the relatiisti additio of eloities is applied wheeer we utilize the eet eloity measuremet u as measured by the iertial frame K. What is the eloity of the eet with respet to the iertial frame K as measured by the rest frame K? Sie both u ad are both kow measuremets performed by the rest frame K, the we a simply make the followig dedutio. u u u 1 u u u 1 10

11 I aordae with the rest frame K, u must idiated the resultig eet eloity with respet to the iertial frame K. It beomes immediately lear that the Galilea additio of eloities is a alid additio of eloities wheeer all measuremets are with respet to the rest frame K. Ay theory of mehais utilized aot erase this fat. It is oly whe we allow usage of a eloity ariable that is measured by aother referee frame do we resort to a modified additio of eloities i aordae with the perspetie theory of mehais proposed. The Galilea additio of eloities is still a alid equatio of mehais ad must be adhered to wheeer all ariables are measured by the same referee frame. z z u, w y y K K Figure 6 As the preedig figure suggests, the measuremet of the eet eloity by the iertial frame K is desigated as u, while the measuremet of the eet eloity by the rest frame K is desigated as w. Therefore, the followig equatio must desribe this seario. u w u I-10 1 u w 11

12 As a be see, the rest frame measuremet u is foud utilizig the Galilea additio of eloities. I-A Galilea additio of eloities formulae is required wheeer all measuremets are made by a sigle referee frame. I-B Modified additio of eloities formulae is required wheeer measuremets are a miture of two or more referee frames. III. Relatiisti Aalysis We a ow perform a relatiisti aalysis of the eloity additio formula based upo the preious fidigs. We begi this aalysis with what was foud i the preious setio. u u u 1 u u u 1 u 1 u u u u 1 1 u u 1 u u 1 u u 11 u u 1 1

13 u u u u 1 1 u u u 1 u I-11 u u 1 Therefore, the resultig eet eloity w with respet to the iertial frame K as measured by the rest frame K the beomes as follows. u I-1 w u u 1 At this poit we a be ery speifi. Sie the rest frame K diretly measures w ad, the the resultig eloity w must be the measuremet of the eet eloity with respet to the iertial frame K as measured by the rest frame K. We a further aalyze this relatio with respet to the Loretz trasformatios as follows. u t Now we a perform the required substitutios. w t 1 t w t t w 13

14 t w I aordae with the speial theory of relatiity, the right had side of the preious equatio is the time t measured by a lok i the rest frame K. t t w We also kow that the eloity w is diretly measured by the rest frame K makig the followig equatio readily apparet. I-13 t t w Based upo this relatio, we a olude that i aordae with the rest frame K, the followig relatioship holds true. I-14 wt It is obious that this must be the trasformatio of the -ais with respet to the rest frame K. I aordae with the rest frame K, the followig must also be true. u t I-15 u t Based upo these equatios, the followig relatioship must hold true i aordae with the rest frame K. z z u, w y y K K Figure 7 14

15 I order to erify this relatioship, we merely retur to the Galilea additio of eloities as gie by the rest frame measuremets. u w, u, w t t t t I-16 t We kow this equatio to be the liear sum of legths as determied by the rest frame K. z z t y y K K Figure 8 15 What this aalysis shows is how the Loretz otratio is treated withi the Loretz trasformatios. We kow that the Loretz otratio is refereed by Eistei i Relatiity: The Speial ad Geeral Theory therefore legitimizig this effet withi his theory. Based upo the preious equatio, we a oly olude the followig. This relatio idiates that this distae must i atuality Loretz Cotratio represet the Loretz otrated distae, ad that the distae must represet the distae that the eet would traerse if it origiated i the rest frame K. We a further derie this relatio by startig with the first equatio of the Loretz trasformatios.

16 t t t Based upo the kow ad obsered mehais of motio, we a make the followig dedutios. t t Sie the origial ierse trasformatio for legth was based upo the trasformatios for time, the we must suspet the temporal Loretz trasformatios as beig the soure of this dilemma. The appedi shows the reliae upo the temporal trasformatio i deduig the ierse Loretz trasformatios. We must eer aept the ommo substitutio gie i some tets as follows., t t, This method of deriatio is lazy ad deries from assumptio rather tha fat. 16

17 Appedi A The Loretz trasformatios are gie by Eistei as follows. t y y z z t t The ierse equatios are obtaied by the followig method. First, we start with solig for the temporal alue. t t t t t A-1 t Now we proeed with resolig the distae. t t t t, t t t t t 1 17

18 t A- t Now we resole the temporal alue. t t, t t t t t t t t t t t t1 t t t t t A-3 t t Therefore, we hae the followig Loretz Trasformatios (LT) ad Ierse Loretz Trasformatios (LT -1 ) as gie by Eistei. t t y y y y 1 A-4 LT zz z z LT t t t t 18

19 1. Eistei, A., O the Eletrodyamis of Moig Bodies, Aale der Physik, 1905, 17: Eistei, A, Relatiity: The Speial ad Geeral Theory, 1916, Methue & Co Ltd. 3. Laue, M., The Etraimet of Light by Moig Bodies Aordig to the Priiple of Relatiity. Aale der Physik, 1907, 3: Fresel, A., Lettre d'augusti Fresel a Fraois Arago sur l'ifluee du mouemet terrestre das quelques pheomees d'optique, Aales de himie et de physique, 1818, 9: Mihelso, A., Ifluee of Motio of the Medium o the Veloity of Light, Ameria Joural of Siee, 3 rd, Vol. 31, No. 185, May Fizeau, M., Sur les hypotheses relaties a l'ether lumieu, Comptes Redus, 1851, 33: Fizeau, M., O the Effet of the Motio of a Body upo the Veloity with whih it is traersed by Light, Philosophial Magazie ad Joural of Siee [ Fourth Series ], Lodo, Ediburgh, ad Dubli, April Hoek, M., Verslage e Mededeelige der Koikl, Akademy a Weieshappe, 1868, d Series, T, II,page Reshaw, C., The Eperimet of Fizeau as a Test of Relatie Simultaeity, 1998, Retrieed February 11, 011 from 19