Experimental evidence of the ether-dragging hypothesis in GPS data

Transcription

1 Experimenal evidence of he eher-dragging hypohesis in S daa Masanori Sao Honda Elecronics Co., Ld., Oyamazuka, Oiwa-cho, Toyohashi, Aichi , Japan Absrac In global posiioning sysem (S) saellies, he earh-cenered locally inerial (ECI) coordinae sysem is used for calculaions. We canno use oher reference frames, for example one based on he solar sysem, in he calculaion of S saellies, because if he relaive velociies in he solar sysem are used, large periodic orbial deviaions of reference ime are calculaed. Therefore, he ECI coordinae sysem is a saionary graviaional frame. This fac provides experimenal evidence for he eher-dragging hypohesis in which he eher is assumed o be he permiiviy of free space, ε, and he permeabiliy of free space, µ. This is inerpreed using he analogy of an acousic wave ha is raveling in he amosphere which is dragged by he graviy of he earh. ACS number: 3.3.+p Key words: S, ECI coordinae sysem, Lorenz ransformaion, eher-dragging hypohesis 1. Inroducion The global posiioning 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 v () km/s. Therefore, he difference in graviaional poenial beween he ground and he locaion of he S saellies and he ransverse Doppler shif effec of special relaiviy on he moion of he S saellies are considered. The ransverse Doppler shif, or second-order Doppler shif, can be calculaed by he Lorenz ransformaion of reference ime. In S, he earh-cenered locally inerial (ECI) coordinae sysem is used for he calculaion of reference ime. The ECI coordinae sysem is fixed o earh's cener; hus, he earh is roaing as shown in Fig. 1 (a). The S uses he relaive velociies defined in he ECI coordinae sysem. Table 1 shows a summary of he S experimen from Ashby [1]. This discussion is carried ou using he Lorenz ransformaion for he calculaions. 1

2 Table 1 Summary of he S experimen from Ashby [1] Term Time difference Condiion raviaional poenial 45.7 µs ime gain every day Heigh:, km from ground level Velociy 7.1 µs ime delay every day v () km/s Eher and explanaions of graviaion were discussed from he 16 h o 19 h cenuries by many of he greaes scieniss, for example, Newon, Maxwell, and Lorenz. From he equaion of he phase velociy of elecromagneic wave, c ε 1 µ, we assume ha he eher is he permiiviy of free space, ε, and he permeabiliy of free space, µ. (ε is he elecric permiiviy of he eher in farads per meer; µ is he magneic permeabiliy of he eher in henrys per meer) Thereafer, he eher is dragged by he graviaional field of he earh. This classic idea was commonly disseminaed and discussed in he days of he 19h cenury. The aberraion of ligh was observed by Bradley in 175. He explained he aberraion using Newon s paricle propery of phoons, producing a simple illusraion of a phoon raveling in a sraigh line in he moving eher, wihou changing is direcion. The aberraion was considered o be one of he experimenal resuls ha show here is no eher-dragging around earh. This aberraion is difficul o explain using he wave naure of he phoon; however, i is easily explained using he paricle naure of he phoon. Therefore, he aberraion does no rule ou eher-dragging. Michelson and Morley [] denied he saionary eher. In he early h cenury, i was hypohesized ha eher-dragging occurs on he ground level. To check his hypohesis experimenally, he Michelson-Morley experimen was carried ou using massive lead blocks (one pah of he inerferomeer was se beween wo lead blocks); here was no fringe shif [3]. Today, he S experimens show ha here is no saionary eher a leas up o, km, because he S works well a orbis of, km in heigh. If here is eher drif, i will be observed as an eher-wind more han, km from he ground level. In 1933, Miller [4] repored experimenal daa ha showed a sligh seasonal and sidereal periodic fringe shif in he Michelson-Morley experimen. However, in 1955, his experimenal resuls were re-evaluaed and found o be hermal arifacs [5]. I believe ha Miller s experimenal resuls showed ha he inerferomeer measuremens are affeced by he moion of he earh. I will discuss he eher-dragging hypohesis and he Michelson-Morley experimen in secion 5. In 19, a he Universiy of Leyden, Einsein described he eher in he heory of general relaiviy; he used graviaional eher in English ranslaion [6]. I was a surprise for me o know Einsein emphasized he imporance of he eher in he heory of general relaiviy. I consider ha his was he firs suggesion of applying he concep of eher o graviy wih respec o he heory of

3 general relaiviy. In 1951, Dirac [7] referred o he eher in he conex of his new elecromagneic heory. He suggesed describing he eher from he viewpoin of quanum mechanics, ha is, quanizaion of he eher. Dirac s suggesion is imporan o connec he heory of relaiviy and quanum mechanics. Finally, using simple numerical calculaions, we will be able o predic he periodic orbial reference ime deviaion in S saellies if he solar sysem is inroduced [8]. 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 solar sysem has no been deeced [1]. Therefore, only he ECI coordinae sysem is correc. Tha is, he ECI coordinae sysem is a saionary reference frame. To explain his conclusion, one possible soluion based on he eher-dragging hypohesis [9] is proposed where, he eher-dragging heigh is more han, km from ground level.. Consideraion of he ECI coordinae sysem The ime of he S saellie is calculaed based on he ECI coordinae sysem and works well. Why does he inerial sysem of he ECI coordinae sysem operae well? Tweny four saellies are launched on six orbis. I is supposed ha he earh is moving a 3 km/s in he solar sysem, and he relaive velociy of he earh in he solar sysem is se o v E. The discussions are carried ou in he solar sysem, hus, he velociies are relaive o he solar sysem. 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, as shown in Fig. 1 (a). Time dilaion in he ECI coordinae sysem is calculaed as follows: r v c 3, (1) c v , () where is he reference ime, is ha of he S saellie, v () km/s is he velociy of he S saellie defined in he ECI coordinae sysem, and is defined as he ime on earh eliminaing he graviaional effec and earh s roaional effec. A ime delay is accumulaed such ha in 1 hour, he deviaion of he car navigaion sysem is roughly esimaed o be 3, , 1 3 m 6 m. The accumulaion of ime dilaion occurs because only he clock in he S saellie suffers he Lorenz ransformaion. This shows ha he 3

4 S is a very sensiive experimenal seup for a ime dilaion measuremen. 3. Calculaion in he solar sysem This secion shows ha he relaive velociy defined in he solar sysem canno be applied o he S calculaions. A simple calculaion shows ha we can only use he relaive velociy defined in he ECI coordinae sysem. The purpose of he numerical calculaion is o show ha here is a difference beween he condiions in he graviaional poenial of he earh and free space. Le us consider he S saellies moving in free space. Figure 1 (b) shows he illusraion of he calculaion in free space. The S saellies orbi in he solar sysem, where here is no graviy and herefore no cenral force from he earh. The S saellies are conneced by ligh rigid arms wih he roaion cener. In he S experimen in free space, here is no graviy of he earh. Numerical calculaions are carried ou using he model in Fig 1 (b), which shows periodic deviaions. The difference beween Figs. 1 (a) and (b) is he graviy. The model in Fig. 1 (a) is checked experimenally; however, ha in Fig. 1 (b) is a hough experimen. Thus, he calculaions are carried ou in he solar sysem using he relaive velociy of he earh v E and ha of he S saellie v (). The numerical calculaions may defocus he discussion; herefore, I will summarize hem in an appendix. 3.1 Orbis V and The orbi V is shown in Fig., where he moion of he S saellie v () is perpendicular o v E, causing he orbi V o become a spiral rajecory in he solar sysem. The period of v () is 1 hours. The ime dilaion is 7. µs per day. These calculaions are shown in he appendix. 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 v () in he solar sysem is periodically changed every 1 hours. The reference ime is calculaed using equaion (3) from he Lorenz ransformaion by seing v E 3 km/s and v km/s, r r ve + v ( ) c (3) r r +. Equaion (3) shows ha here is a periodic deviaion depending on he velociy ( v v ()) E The periodic derivaion of he reference ime is calculaed as ± ; hese 4

5 calculaions are shown in he appendix. The deviaion is periodic, which causes a deviaion in disance of around.8 km. However, he ECI coordinae sysem operaes well by he S saellies, meaning no orbi-dependen periodic disance deviaion is observed. The deviaion of he reference ime of he orbi is similar o ha of he orbi V. Thus, we canno use he relaive velociy defined r r v + v (). in he solar sysem ( ) E 3. Summary of he calculaion The calculaions in he appendix are simple bu numerous;; herefore, I will summarize hem briefly here. (1) The S is a very sensiive experimenal seup o check he ime dilaion. () S calculaions canno be carried ou in he solar sysem using he relaive velociy of he earh v E and ha of he S saellie v (). (3) A simple explanaion for he ECI coordinae sysem is required. The reason for which he S works well in he ECI coordinae sysem bu no in he solar sysem is discussed in Secion 4 as a proposed soluion. (4) The difference beween Figs. 1 (a) and (b) is graviy. The model in Fig. 1 (a) has been checked experimenally. Alhough he model shown in Fig. 1 (b) was derived from a hough experimen, he numerical calculaion based on his model seems o be correc. 4. roposed soluion Since he deviaion of he reference ime mus be in agreemen wih he resuls of he ECI coordinae sysem, he calculaion using he ECI coordinae sysem includes no only he effecs of special relaiviy, bu also of general relaiviy. However, a simple argumen shows ha i is almos impossible o calculae he S in he solar sysem. One of he possible soluions is ha he graviaional field of he earh is a saionary graviaional field. This is derived from he eher-dragging hypohesis in which he graviaional field of he earh drags he eher, where he eher is he permiiviy of free space, ε, and he permeabiliy of free space, µ. The S works in he eher dragged by he graviaional field of he earh [9]. Figure 4 shows he eher dragging model wih he ECI coordinae sysem, he solar sysem, he galaxy, and he CMB are respecively in local saionary ehers. This is because each graviaional field drags he ehers around is graviaional field. The galaxy moves in he CMS a 7 km/s, he solar sysem moves in he galaxy a 3 km/s, and he ECI coordinae sysem moves in he solar sysem a 3 km/s, corresponding o poins a o d in Fig. 4 in local saionary saes. The S saellie in he ECI coordinae sysem observes 4 km/s, bu i does no deec he relaive velociy in oher coordinaes. Thus, he S saellie is in he saionary sae of he ECI coordinae sysem wih relaive velociy 4 5

6 km/s. If he S saellie leaves he ECI coordinae sysem from poin a o poin b, he local saionary sae for he S saellie is changed o he solar sysem from he ECI coordinae sysem. The S saellie ha moves parallel o he earh observes he velociy 3 km/s in he solar sysem. If we reach he graviaional field of Mars, we will be in anoher saionary graviaional sae, namely he Mars-cenered locally inerial coordinae sysem. There may be anoher soluion derived from he calculaion using boh special relaiviy and general relaiviy; however his calculaion is very complicaed. In his case, i becomes acceleraed moion and he discussion using general relaiviy is required. The orbi of he S saellie and he orbi of he earh seen from arbirary inerial sysems are shown in Fig. 5. The orbi is a cycloid, so acceleraion and deceleraion are repeaed. I have no idea how o carry ou he calculaion in an arbirary reference frame. Therefore, I choose he eher-dragging hypohesis, which provides a simple calculaion for he S experimens. A he disance measuremen of, km (his is a disance beween he car and he S saellie), he ime deviaion 9 ± causes a 6 mm deviaion. (,km mm ) I is raher small; however, he deviaion is accumulaed for 6 hours. Le he value of he average deviaion be half of he maximum deviaion 9 ± Thus, we obain, muliplying a rough esimaion of ½ he averaged deviaion ½, 1 9,km , km. (4) This calculaion shows ha if he clock on he S saellie is lef alone wihou an adjusmen for 6 hours, he accumulaed deviaion of he disance is roughly.8 km. In he S, no such deviaion is observed. 5. Discussion The auhor hanks he reviewers for imporan commens on he earh's eccenriciy as well as he suggesion ha he discussion should no largely depend on he numerical calculaions. From he viewpoin of he eher-dragging hypohesis, he Sagnac effecs, he ECI coordinae sysem, he aberraion, he Michelson-Morley experimens, and he analogy of an acousic wave are discussed. 5.1 Earh's eccenriciy A perihelion he disance from he earh and he sun is.983 AU (asronomical uni: average disance beween he earh and sun, m), ha of aphelion is 1.17 AU. Thus he velociy r deviaion is esimaed o be.5 km/s, ( v E ( ) 3 ±.5km / s), which causes abou 1.6 mm of deviaion a he measuremen of, km, 6

7 ,,km 1. 6mm. (5) 3 3, The period of deviaion is 1 monhs. The reference ime deviaion is no accumulaed; his is because no only he clocks on he S saellie bu also he clocks on earh simulaneously differ according o v E. Therefore, here is no accumulaion of he reference ime deviaions. Thus, he deviaion relaes o one ime measuremen of disance, for example,, km is a disance beween he S and a car on earh; only 1.6 mm is calculaed as he deviaion. However, a he velociy of v km/s, only he clock on he S saellie obains he deviaion; hus, he deviaion is accumulaed for half he period of revoluion as shown in equaion (4). The accumulaion causes a difference of beween.8 km and 1.6 mm. In he inerferomeer of he L1 band ( MHz, wavelengh: 19 mm), relaive accuracies of millimeers are repored [1]. This is he precise posiioning analysis wih carrier-phase measuremens. However, a his sage, he resoluion is 1% of he wavelengh ( 19 mm 1/1 1. 9mm ), hus he value of 1.6 mm is less han he echnical limi of he measuremen. However, he deviaion of 1.6 mm is no observed for heoreical reasons. We canno deec any annual deviaions dependen on he velociy of he earh. In his discussion, I assumed ha he S is in he dragged eher, so we canno deec any deviaions. Even if advanced carrier-phase measuremens wih relaive accuracies of sub millimeer are used, i is impossible o deec a 1.6 mm deviaion ha depends on he earh's eccenriciy. The S experimens show ha we canno observe he reference ime deviaions represened by equaions (3) and (5). 5. Sagnac effec Sagnac effecs reflec a changing disance beween he ligh source and he observer caused by moion of he observer. If he observer moves during he fligh ime of he ligh, he disance beween he ligh source and he observer is changed. For example, le he disance beween he S saellie (signal source) and he observer on earh be 3, km (he disance ha ligh ravels a 1 second). On he equaor, he speed of he ground is abou.47 km/s. Thus he Sagnac effec is.47 km a he measuremen of 3, km for he observer on he equaor, meaning ha he fligh ime of ligh of 1 second, he observer moves.47 km. According o Ashby [1], Sagnac effecs are experimenally observed wihin a % deviaion. Therefore, he deviaion on he measuremen of he Sagnac effec is calculaed o be.47 km. 3,km This value is equivalen o he deviaion of he eher roaion in he ECI coordinae sysem. In oher words, he eher is almos fixed 7

8 o he ECI coordinae sysem. Thus, he eher has wo properies as shown in Fig. 6: 1) he eher is dragged wih he earh, ) he eher does no roae wih he earh; if he eher is fixed o he ECI coordinae sysem, he deviaion of he roaion is roughly esimaed o be less han The angular frequency of he earh s roaion in he ECI coordinae sysem is rad s 5 ω E [1], and he period is around 3 hours 56 minues and 4.1 seconds. Thus, I assume ha he earh roaes in he eher a he angular frequency ω E. 5.3 roposed experimen of local posiioning sysem To explain he eher-dragging hypohesis, I describe oher discussions of local posiioning sysems. The orbial cener of he S saellies is he cener of he earh, and he S saellies have symmerical orbis and he velociy v km/s in he ECI coordinae sysem. Le us consider hree geosaionary saellies lying over he equaor a km, 6,4 km, and 1,8 km from he ground. The relaive velociies in he ECI coordinae sysem are.47 km/s,.94 km/s, and 1.41 km/s, respecively. Therefore, hree geosaionary saellies observe ime dilaions ha depend on heir velociies. The model in Fig 1 (b) can be checked around he earh in local posiioning sysems as shown in Fig. 7. If his local posiioning sysem is se around he geosaionary saellie, he relaive velociies of he saellies defined in he ECI coordinae sysem have orbial deviaions, and here are deviaions of he reference ime. Numerical calculaions will show ha he local posiioning sysem needs raher complicaed calculaions o work well; in paricular, we have o modify he calculaion equaions o use his sysem. If he proposed experimen is carried ou, we may possibly obain addiional experimenal evidence of he eher-dragging hypohesis. Figure 1 (b) shows if here is no graviaional field of he earh, he ECI coordinae sysem is no defined. The origin of he ECI coordinae sysem is he earh; he ECI coordinae sysem is defined by he graviaional field of he earh. Therefore, he earh generaes he ECI coordinae sysem. 5.4 Saionary sae Wha makes he ECI coordinae sysem and he saionary sae? The answer is he graviaional field of he earh. The ECI coordinae sysem canno be defined wihou he graviaional field of he earh. Figure 8 shows he earh and he local posiioning sysem, he saionary sae in he ECI coordinae sysem is also illusraed. The cener saellie of he local posiioning sysem is saionary in he ECI coordinae sysem, and we shall call i he ECI saionary saellie. The origin ha defines he ECI coordinae sysem is he earh; he origin of he saionary sae in he ECI coordinae sysem is he earh. If his local posiioning sysem is se away from he 8

9 graviaional field of he earh, he ECI saionary saellie experiences he relaive velociy of 3 km/s in he solar sysem. The graviaional field of he earh generaes he ECI coordinae sysem and he saionary sae, in which ime passes mos rapidly. The ime dilaion occurs according o he relaive velociies defined in he ECI coordinae sysem, which can be plausibly explained using he eher and eher-dragging hypoheses. 5.5 Aberraion The aberraion is a counerargumen agains he eher-dragging hypohesis. If ligh is a wave in he eher, he ligh is dragged by he eher and he aberraion canno be observed on he earh. I is said ha he aberraion canno be compaible wih eher-dragging, bu Bradley s explanaion using Newon s paricle model of ligh shows he compaibiliy of aberraion and he eher-dragging hypohesis. Figure 9 shows aberraion and eher-dragging. The wave-paricle dualiy shows ha a phoon ravels perpendicular o he wave fron, which is also perpendicular in he dragged eher by he earh. The relaive velociy beween he saionary eher in he solar sysem and he dragged eher is 3 km/s. The boundary beween he saionary eher and dragged eher is more han, km from ground level. hoons ravel in sraigh lines in he solar sysem and dragged eher. According o quanum mechanics, he phase velociy, c, of a phoon is defined as cω/κ (ω: frequency, κ: wave number); i is also he raio of he energy ε and momenum µ of he phoon, cε/µ. The energy and momenum are conserved o saisfy he consancy of he speed of ligh, c. Therefore a simple example of a phoon raveling in a sraigh line in he moving eher wihou changing is direcion is possible. In he aberraion, a phoon is deeced as a paricle, and i is correc o use he paricle properies of phoons. Alhough i is impossible o explain he aberraion by he wave propery of phoon, I do no consider ha he aberraion rules ou eher-dragging. 5.6 Michelson-Morley experimen and S The difference beween he Michelson-Morley experimens and S experimens is he following: 1) he Michelson-Morley experimens are carried ou in he roaing frame in he ECI coordinae sysem, ) he S experimens are done in he ECI coordinae sysem. Figure 6 shows he roaing frame (surface of he earh) and he ECI coordinae sysem. Figure 9 shows ha around he ground level on he equaor, he relaive velociy beween he ground and dragged eher is.47 km/s. Thus, as described previously, he fringe shif observed in he Michelson-Morley experimens depends on he velociy of.47 km/s. I is raher difficul o observe he roaion of he earh wih he Michelson inerferomeer, bu he S can observe he roaion of he earh as a Sagnac effec a he 9

10 sensiiviy of % [1], or a velociy of 9.4 m/s. The S shows he isoropic consancy of he speed of ligh. The disance beween he S saellie and he car is calculaed using he ime delay D as 3, km / s D ( s), where he speed of ligh c3, km/s is assumed o be an isoropic consan. Therefore, he fac ha S works well is evidence for he isoropic consancy of he speed of ligh. The sensiiviy of a direc 4 one way measuremen (from he S saellie o a car on earh) in S is 1 higher compared o he Michelson inerferomeer. Thus he null resuls are confirmed by he S. Michelson and Morley [] repored ha he relaive velociy of he earh and he eher is probably less han one sixh he earh orbial velociy (5 km/s) and cerainly less han one fourh (7.5km/s). No only Miller bu also Michelson and ohers have repored he eher drifs below 1 km/s. Alhough, he deviaions in he Michelson-Morley experimenal resuls were considered as hermal arifacs [5], hese deviaions are criical evidence of he counerargumen agains eher-dragging. Thus, i is imporan o sudy he source of he deviaions. In 1887, Michelson and Morley [] assumed he earh s revoluion in he solar sysem was 3 km/s. In 1933, Miller [4] assumed he solar sysem moion in he galaxy o be km/s in addiion o he earh s revoluion in he solar sysem of 3 km/s. I consider ha according o eher-dragging, null resuls are prediced; however, he experimenal deviaions observed by Miller and Michelson are large enough no o be negligible. I believe hese experimenal resuls of periodic deviaions show ha he inerferomeer measuremens are affeced by he roaion of he earh. 5.7 Analogy of acousic wave To make he discussion more clear, le us consider he analogy of an acousic wave in he amosphere, which is compleely dragged by he graviy of he earh. Thus he moion of he earh in he solar sysem does no affec he speed of he acousic wave. From his analogy, I derived he eher-dragging hypohesis. The difference beween he eher and he amosphere is roaion. The amosphere roaes 5 synchronously wih he earh wih he angular frequency is ω E rad s ; on he oher hand, he eher does no roae (ha is, ω ). A he beginning of he paper, from he analogy of an acousic wave, I considered ha he eher is fixed o he amosphere and roaes synchronously wih he earh. However, according o he S experimens, I hink he eher is dragged bu does no roae; a his sage, I canno explain he reason. The S saellie is moving in he ECI coordinae sysem, in he solar sysem, in he galaxy, and in he CMB. However, he reference ime of he S is only affeced by he relaive velociy defined in he ECI coordinae sysem. Numerical calculaion shows ha we canno use he S in he solar sysem; his is one of he experimenal pieces evidence for he eher-dragging hypohesis. E 1

11 5.8 Summary of eher-dragging hypohesis (1) The eher is he permiiviy of free space, ε, and he permeabiliy of free space, µ. () The eher is dragged by he graviaional field of he earh. (3) The eher does no roae wih he earh; he eher is fixed o he ECI coordinae sysem. (4) The eher-dragging hypohesis has been improving o be compaible wih almos all he hisorical experimens as well as he S experimens. (5) A his sage, we have o use he wave-paricle dualiy in he explanaion of he compaibiliy wih he aberraion. 6. Conclusion In he S calculaions, we can only use he ECI coordinae sysem; anoher reference frame, for example one based on he solar sysem, canno be applied o S experimens. Therefore, he graviaional field of he earh is he saionary graviaional field, and he ECI coordinae sysem works well. This is he experimenal evidence supporing he eher-dragging hypohesis. This hypohesis is inerpreed using he analogy of an acousic wave ha is raveling in he amosphere ha is dragged by he graviy of he earh. I consider ha we should reexamine he classical eher and eher-dragging hypohesis, which should have a compaibiliy wih he heory of relaiviy, quanum mechanics, S experimens, and space physics. References 1) N. Ashby, Relaiviy in he lobal osiioning Sysem, (3). ) A. Michelson, and E. Morley, "On he Relaive Moion of he Earh and he Luminiferous Eher," American Journal of Science, Third Series, 34, 333, (1887), hp:// 3). Hammar, "The Velociy of Ligh wihin a Massive Enclosure," hysical Review 48, 46, (1935). 4) D. Miller, "The Eher-Drif Experimen and he Deerminaion of he Absolue Moion of he Earh", Reviews of Modern hysics, 5, 3, (1933). 5) R. Shankland, S. McCuskey, F. Leone, and. Kueri, "New Analysis of he Inerferomeer Observaions of Dayon C. Miller," Rev. Mod. hys., 7, 167, (1955). 6) A. Einsein, "Eher and he Theory of Relaiviy" (19), republished in Sidelighs on Relaiviy, (Dover, NY, 1983). 7). Dirac "Is here an eher?" Naure, 168, 96, (1951). 8) M. Sao, "Inerpreaion of special relaiviy as applied o earh-cenered locally inerial 11

12 coordinae sysems in lobal osiioning Sysem saellie experimens," arxiv:physics/57v4, (6). 9) M. Sao, A revisi of he papers on he heory of relaiviy: Reconsideraion of he hypohesis of eher-dragging, arxiv:74.194v5, (9). Appendix: Numerical calculaions in free space (1) Orbi V 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 in he ECI coordinae sysem is se o v. When he reference ime of he reference frame a res is se o and ha of he roaion cener is se o E and is expressed wih, he Lorenz ransformaion is as follows: E r ve 3 c 3, E ,. (A-1) The moion of a S saellie is perpendicular o ha of he roaion cener such ha v r r E v. Because v r r E v, we can use he yhagorean proposiion o obain he summaion of v E and v and he reference ime V of he S saellie by he following equaion: V r ve + c r v , V , (A-) Therefore, he proporion of he ime delay over he roaion cener of he S saellie is as follows: 1

13 V E E V E 3 3, , (A-3) The difference beween equaions () and (A-3) appears only afer he 8h figure. Thus we obain, µs, and i becomes he delay of 7. 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 equaion (A-) and (A-3) we se v km/s). () Orbi Nex, he orbi is considered. In Fig. 3, he summaion of he velociy of he roaion cener v E and he velociy of he S saellie in he solar sysem is periodically changed. A periodic derivaion of he reference ime is expeced; for example, he reference ime is calculaed using equaion (A-4) from he heory of special relaiviy by seing v E 3 km/s and v km/s. r r ve + v ( ) c (A-4) 1 r r ve + v c ( ). (A-5) 1) A poin A in Fig. 3, where v r r // E v, ,. (A-6) r r ) A poin C in Fig. 3, where v E // v, 13

14 ,. (A-7) 3) A poins B and D in Fig. 3, where v r r E v, r r ve + v c ( ) (A-8) , (A-9) The differences beween equaions (A-9) and (A-6), (A-7) is calculaed as 9 ± 1.3 1, where is he reference ime of he S saellie when v E and v are parallel, and he deviaion is calculaed o be ± However, he deviaion becomes one order larger compared o he value obained from equaion (1). The deviaion is sinusoidal; he periodic deviaion is esimaed o be more han 1 km. The ECI coordinae sysem operaes well by he S saellies, and 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. 14

15 Orbi V S saellie Orbi v E 3 km/s Direcion of earh moion in he solar sysem Fig. 1 (a) The earh-cenered locally inerial (ECI) coordinae sysem. The S saellies and he earh moion in he solar sysem. Orbi is parallel and orbi V is perpendicular o he direcion of earh moion in he solar sysem, v E 3 km/s. The ECI coordinae sysem is fixed o earh's cener; hus, he earh is roaing. The S saellies orbi a he relaive velociy v () km/s defined in he ECI coordinae sysem. If he S saellies are seen from he solar sysem, heir orbis are cycloid. However, he deviaion of he S saellies is no observed. Time dilaion only depend on he relaive velociy v () km/s. Orbi V S saellie Roaion cener Orbi v E 3 km/s Direcion of he frame moion in he solar sysem Fig. 1 (b) Though experimen of he S in free space in he solar sysem: here is no graviy of he earh, S saellies and roaional cener are combined by ligh rigid lodes. Numerical calculaions are carried ou using his model, which shows periodic deviaions. The difference beween Figs. 1 (a) and (b) is graviy. The model in Fig. 1 (a) is checked experimenally; however, ha in Fig. 1 (b) is hough experimen, where he relaive velociies are defined in he solar sysem. 15

16 v 4 km/s S saellie v E 3 km/s Roaion cener C Orbi V Fig. Orbi of saellie moving perpendicular o he direcion of earh moion in he solar sysem. Orbi V: The orbial plane of he S saellie is perpendicular o he direcion of he earh moion in he solar sysem. D Roaion cener B v 4 km/s v E 3 km/s S saellie Orbi A Fig. 3 Orbi of saellie moving parallel o he direcion of earh moion in he solar sysem (orbi ). The velociy of S saellie in orbi has an orbial deviaion. However, his illusraion is no compaible wih he S experimenal daa. There are no periodic orbial deviaions. Cosmic microwave background d v7 km/s c Solar sysem Sun a b alaxy v3 km/s ECI coordinae sysem v3 km/s Fig. 4 Eher dragging model: he ECI coordinae sysem, he solar sysem, and he CMB are respecively in he local saionary ehers. This is because each graviaional field drags he ehers around is graviaional field. There are many graviaional fields, hus, here are many local saionary saes. Each poins a o d are in local saionary saes. If he S saellie leaves he ECI coordinae sysem, i will be ino he local saionary sae of he solar sysem. 16

17 Deceleraion Acceleraion Deceleraion Fig. 5 Traveling pah of he S saellie of orbi in an arbirary reference frame. From an arbirary reference frame, saellie moion is periodic, orbial acceleraed moion of cycloid. v E 3 km/s Dragged eher Fig. 6 Eher-dragging model: 1) he eher is dragged wih he earh, ) he eher does no roae wih he earh; he eher is fixed o he ECI coordinae sysem. Fig. 7 Local posiioning sysem: his sysem orbis around he geosaionary saellies lie over he equaor. This proposed experimen shows ha he graviaional field of he earh generaes he ECI coordinae sysem. 17

18 Local posiioning sysem Earh ECI saionary saellie ECI coordinae sysem Fig. 8 Saionary sae in he ECI coordinae sysem: The cener saellie of he local posiioning sysem is saionary in he ECI coordinae sysem, le call i he ECI saionary saellie. The origin of he ECI saionary saellie is he earh. If his local posiioning sysem is se ou of he graviaional field of he earh, i suffers he relaive velociy of 3 km/s in he solar sysem. Wave fron Saionary eher in he solar sysem Dragged eher, km v E 3 km/s Earh Fig. 9 Aberraion and eher-dragging: wave-paricle dualiy shows ha a phoon ravels perpendicular o he wave fron which is also perpendicular in he dragged eher by he earh. Bradley s explanaion using Newon s paricle model of ligh shows he compaibiliy of aberraion and eher-dragging hypohesis. The relaive velociy beween he dragged eher and he saionary eher in he solar sysem is 3 km/s. The boundary beween he saionary eher and dragged eher is more han, km from ground level. hoons ravel on he sraigh line in he solar sysem and dragged eher. 18