Interpretation of special relativity as applied to earth-centered locally inertial

Transcription

1 Inerpreaion of special relaiviy as applied o earh-cenered locally inerial coordinae sysems in lobal osiioning Sysem saellie experimens Masanori Sao Honda Elecronics Co., Ld., Oyamazuka, Oiwa-cho, Toyohashi, Aichi , Japan msao@honda-el.co.jp Absrac In he lobal osiioning Sysem (S) saellies, eiher he earh-cenered, earh-fixed, reference frame (ECEF frame) or he earh-cenered locally inerial (ECI) coordinae sysem is used for calculaions. In he applicaion of he heory of special relaiviy o S saellies, we have o use he ECEF frame, which is pracically a reference frame which is a res. We poin ou ha anoher reference frame a res ( for example, one based on he cosmic microwave background or he solar sysem) can be applied o S saellies experimens, however, he calculaion needs no only special relaiviy, bu also general relaiviy. ACS number: 3.3.+p Key words: special relaiviy, reference frame a res, Lorenz ransformaion, S, ECEF frame, ECI coordinae sysem, 1. Inroducion The heory of relaiviy is used in our daily lives. The lobal osiioning Sysem (S) is used in car navigaion sysems. The use of special relaiviy in S has been summarized by Ashby [1]. The S saellies orbi in a region of low graviy (~, km from ground level) a 4 km/s. Therefore, he difference in graviaional poenial beween he ground and he S saellies, and he effec of special relaiviy on he moion of he S saellies, which is a ransverse Doppler shif, are considered. The ransverse Doppler shif, or second-order Doppler shif, can be calculaed by he Lorenz ransformaion of ime. In S, he earh-cenered, earh-fixed, reference frame (ECEF frame) or he earh-cenered locally inerial (ECI) coordinae sysem is used for he calculaion. The ECEF frame is pracically a reference frame a res (a saionary reference frame). Twin paradox experimens are carried ou every day in he S saellies. The difference in graviaional poenial beween he S saellies and he ground causes a 45.7 µs ime gain every day. The ransverse Doppler shif ha can be calculaed by he Lorenz ransformaion of ime resuls in a 7.1 µs ime delay every day. We resric his discussion o wihin he special relaiviy heory, ha is, we do no consider he effec of he graviaional poenial. Thus, he raveling win is an aomic clock in a S saellie, and he saionary win is an aomic clock on earh. The raveling win becomes anoher 7.1 µs younger han he win on earh every day. As menioned above, one of a pair of wins who reurns from a space rip will be younger han he oher one 1

2 who remained on earh. However, from he viewpoin of relaiviy, he win who has reurned from he space rip can claim ha he oher win who sayed on earh moved away from he space ship: his is he win paradox. Twin paradox experimens have been carried ou in S saellies; he experimenal resuls show ha we see old and young aomic clocks. The resuls prediced by he heory of special relaiviy are correc, however, he inerpreaion of relaiviy gives rise o he paradox. Tha is, we can claim ha he clock in he S saellie underwen moion relaive o he clock on earh. Of course, he siuaion is differen from he viewpoins of he S saellie and earh because he S saellie undergoes acceleraed moion. I is easy o resolve he paradox if we adop a reference frame a res, while i is raher difficul wihou a reference frame a res. Finally, using simple numerical calculaions, we show ha if an inerial frame a res is inroduced, we will be able o predic he periodic orbial reference ime deviaion in S saellies. However, in he S saellie experimens, such a large periodic orbial deviaion ha criically depends on he moion of he orbial plane of he S saellies in he cosmic microwave background has no been deeced [1]. Therefore, only he ECI coordinae sysem is correc. However, his conclusion looks incompaible wih he orhodox inerpreaion, ha is, here is no reference frame a res. In his paper, one possible soluion is proposed. Orbi V S saellie Earh Orbi v E 35 km/s Direcion of earh moion in he cosmic microwave background Fig. 1 Earh moion in he cosmic microwave background. Orbi is parallel and orbi V is perpendicular o he direcion of earh moion in he cosmic microwave background. In he cosmic microwave background, v E 35 km/s and in he solar sysem, v E 3 km/s.. Consideraion of he ECEF frame The ime of he S saellie is calculaed based on he ECEF frame and is operaing well. Why does he inerial sysem of he ECEF frame operae well? Tweny four saellies are launched on six orbis and S is operaing well by he ECEF frame. I is supposed ha he earh is moving a 35 km/s in he cosmic microwave background. The relaive velociy of he earh in he cosmic microwave background is se o v E. The cosmic microwave background is assumed o be a reference frame a res, and he ime delay of he S

3 v 4 km/s S saellie Earh v E 35 km/s Orbi V Fig. Orbi V: he orbial plane of he S saellie is perpendicular o he direcion of he earh moion in he cosmic microwave background. saellie is examined on he basis of he reference ime on earh. There are wo ypes of S orbis, one is he orbi ha is parallel o v E, and he oher is he orbi V ha is perpendicular o v E. The orbi and V are shown in Fig.1. The orbi V is shown in Fig., where a moion of he S saellie is perpendicular o v E, hus causing he orbi V o become a spiral rajecory. The velociy of he S saellie relaive o he earh is se o v. When he reference ime of he reference frame a res is se o and ha of he earh is se o E and is expressed wih, he Lorenz ransformaion is as follows, E. (1) Since he moion of a S saellie is perpendicular o ha of he earh, ha is v E v, he reference ime of he S is obained by he following equaion. This is because v E v hus we can use yhagorean proposiion o obain he summaion of v E and v () Therefore, he proporion of he ime delay over he earh of he S saellie is as follows E E (3) Even if i is calculaed in he ECEF frame, i is only afer he 6h figure ha a difference appears, as shown below 3

4 (4) 1 Thus we obain, µs, and i becomes he delay of 7.6 µs per day (There is a difference wih he experimenal daa of 7.1 µs, his difference criically depends on he value of he velociy v, in equaions (3) and (4) we se v 4 km/s). Earh v 4 km/s v E 35 km/s S saellie Orbi Fig. 3 The relaive velociy of S saellie in he cosmic microwave background (orbi ): he relaive velociy of he S saellie in he cosmic microwave background, where he orbial plane of he S saellie parallel o he direcion of he earh moion in he cosmic microwave background has an orbial deviaion of he reference ime. However, he calculaed resul prediced by his illusraion is incompaible wih he experimenal daa, ha is, no periodic deviaion was deeced: we should adop he illusraion in Fig. 4. Nex, he orbi is considered. In Fig. 3, he summaion of he velociy of he earh v E and he velociy of he S saellie in he cosmic microwave background is periodically changed. In his case, i becomes acceleraed moion and he discussion using general relaiviy should be required. The orbi of he S saellie and he orbi of he earh seen from arbirary inerial sysems are shown in Fig. 4. The orbi is a cycloid hus acceleraion and deceleraion are repeaed. The periodic derivaion of he reference ime is expeced; for example, he reference ime is calculaed using equaion (5) only from he heory of special relaiviy by seing v E 35 km/s and v 4 km/s wihou including he effec of he heory of general relaiviy. ve + v ( ) 1 c (5) 4

5 where is he reference ime of he S saellie when v E and v are parallel, and he deviaion is calculaed as 8 ± However, he deviaion becomes wo orders larger comparing wih he value 1 obained from equaion (3). If he solar sysem is assumed o be an inerial sysem, ha is v E 3 km/s, here is a difference of one order larger comparing wih he value obained from equaion (3). However, he ECEF frame operaes well by he S saellies, ha is, no periodic deviaion is observed which depends on he orbis. The deviaion of he reference ime of he orbi is similar o ha of he orbi V. Deceleraion Acceleraion Deceleraion Fig. 4 Traveling pah of he S saellie of orbi in an arbirary reference frame. A clock in he saellie ravels according o he red line. Therefore, a phoon in an aomic clock in he saellie ravels a furher disance han an aomic clock on earh, which causes ime delay in he saellie. From an arbirary reference frame, saellie moion is periodic, orbial acceleraed moion (ha is, cycloid). I conclude ha he calculaion ha includes acceleraion will show he correc resul: any inerial reference frame will give he same resul. Since he deviaion of he reference ime mus be in agreemen wih he resul of he ECEF frame, herefore he calculaion using he ECEF frame is including no only he effecs of he heory of special relaiviy, bu also of general relaiviy. The orbi and V are also considered so ha he same equaion of he reference ime can be used as follows, v 1 E + v c ( ). (6) If he effec of he special relaiviy expressed by equaion (6) is used, i seems ha in any arbirary orbi, he ime delay of he S saellie will become he same. According o Ashby [1], here is no significan periodic deviaion observed. This is considered ha he discussion of equaion (5) does no consider he effec of heory of general relaiviy. If he S saellie is seen from any inerial sysem, he effecs of special relaiviy (ha is, a periodic moion of velociy) and general relaiviy (he effec of acceleraion and deceleraion) offse each oher, herefore equaion (6) can be used. A presen, he effec of he heory of general relaiviy has no been 5

6 calculaed in his repor. A firs, I did no know he reason why he ECEF frame or he ECI coordinae worked well. I considered ha i may be caused by he Lense-Thirring effec, which is an effec ha is produced by a graviaional field of he earh pulls an inerial sysem. This effec was prediced in he early sages of general relaiviy, and was checked in an experimen done in recen years []. I hough ha he graviaional field of he earh had pulled he surrounding inerial sysem by he Lense-Thirring effec. A his sage, I do no consider he S experimenal resuls o have any relaion wih he Lense-Thirring effec. I consider ha eiher he ECEF frame or he ECI coordinae works correcly under he consideraion of he heory of special and general relaiviy. 3. Conclusion We discussed why eiher he ECEF frame or he ECI coordinae works well. We concluded ha any inerial reference frame gives he same ime delay resul for he S saellie experimens. Tha is, he S experimens should be discussed from he viewpoin of general relaiviy (including he effec of acceleraion) as well as special relaiviy (he effec of velociy). References 1) N. Ashby, Relaiviy in he lobal osiioning Sysem, (3). ) I. Ciufolini, and E. avlis, A confirmaion of he general relaivisic predicion of he Lense-Thirring effec, Naure, 431, 958, (4). 6