Propagaion of spherical elecromagneic waves in differen inerial frames; an alernaive o he Lorenz ransformaion Teimuraz Bregadze ebr50@yahoo.com The applicaion of he Lorenz ransformaion, which assumes planar characer of elecromagneic waves, o he poins of he fron of a spherical elecromagneic wave gives a disored picure of he physical realiy. The expressions of Doppler effec and aberraion, as well as he ransformaion law of he wave frons for spherical elecromagneic waves, essenially differen from hose for planar elecromagneic waves, are obained. Some consequences are discussed. Paricularly, he necessiy of inroducion of he naural unis of lengh (changing in he same way as he lenghs of elecromagneic waves) for measuring disances from moving poins o he frons of elecromagneic waves, as well as of he real disance covered by a moving poin relaive o a saionary poin conaining he number of elecromagneic waves which is equal o he difference in he numbers of waves conained beween he fron of an elecromagneic wave and he saionary and moving poins respecively. A revised equaion for he addiion of velociies is also obained. 03.30.+p The equaion of he fron of a spherical elecromagneic wave propagaing from he origin of a saionary frame is ha of a sphere: x + y + z = c, (1) where x, y, z, and are he space and ime coordinaes of a poin of he wave fron, and c is he velociy of ligh in empiness. According o he Lorenz ransformaion, in he frame wih axes coinciding wih hose of he saionary frame a he insan of he sar of emission of he wave (= = 0) and moving wih velociy v = c along axis X, he equaion of he same wave fron is: x + y + z = c, where x c c x x =, y = y, z = z, = () c are he space and ime coordinaes of he same poin of he wave fron in he moving frame. In figure 1 he sphere wih cenre in he origin of he saionary frame, O, represens he wave fron in he saionary frame when he clock a ha poin shows ime ; O 1 is he origin of he moving frame; and OO1 = c. If we ry o represen he same wave fron in he moving frame using equaions (), we will ge a spheroid wih cenre in O and one of he focuses in O 1, where c c O O1 = ; he long half axis of he spheroid is O C = and he shor 1 half axes are equal o O E = c. 1
Indeed, according o equaions (), poin O (,0,0, ) 0 in he saionary frame becomes poin c O,0,0, in he moving frame. If we assume ha he origin of he moving frame is in poin O, hen he equaion of he wave fron in his frame is: c x where x 1 = x + =. ) 1 ( x + y + z = c E E E H A 1 A A A θ α ϕ K X, X D D D 1 O O O 1 O 1 F G C C C 1 B 1 B B Figure 1. The frons of elecromagneic waves in saionary and moving inerial frames.
A he same ime we have o bear in mind ha he ime coordinaes of he poins of he wave fron in he moving frame vary along he axis X from = 1+ a poin C o = a poin D. 1 1+ The design of he Lorenz ransformaion is o give he picure of he physical realiy a moving observer shall have from he poin of view of a saionary observer. Bu he picure ha resuls from his ransformaion is dramaically differen from he one a moving observer really ges in his frame; in oher words, a moving observer never ges he picure of he wave fron we have described above. According o he Lorenz ransformaion, when he moving clock in poin O shows ime =, poin E of he wave fron in he saionary frame becomes poin E in he moving frame. Bu he moving observer acually discovers ha a ha insan of ime in poin O he wave fron is no in poin E, bu in poin E, which is a c poin of he sphere wih cenre in O and radius which he emission of he wave sared in he moving frame is O. Le in poin E of he saionary frame a clock is placed. A he insan when he wave fron arrives a ha poin he clock reads. This fac is an invarian and mus be observed in any frame. According o he Lorenz ransformaion poin E is poin E in he moving frame and he wave fron arrives a ha poin when he clock here shows = R =, because he poin from. Tha is simply impossible: a clock canno show differen imes simulaneously. The Lorenz ransformaion assigns unique qualiies o saionary frames, radically disinguishing hem from he moving ones: ime coordinaes of he poins in saionary frames do no depend on spaial coordinaes and in moving ones hey do. I is unclear how he differences in he readings of he clocks in differen poins arise when a saionary frame sars is movemen. For example, le each carriage of a long rain have is own engine and clock (he carriages, engines and clocks are idenical). When he rain is a res a he saion all clocks on he rain show he same ime as he clocks on he saion. A some insan of ime he engines of all carriages are swiched on simulaneously and he rain sars moving. Why should some ime laer he saionary observers a differen carriages regiser differen differences beween he readings of he moving and heir own clocks? The elecromagneic waves are spherical [1], bu he Lorenz ransformaion requires ha hose waves be planar. This is because he phase of an elecromagneic wave is o be an invarian, and under he Lorenz ransformaion he phase which is he same in any inerial frame is ha of a plane wave []. I is believed ha elecromagneic waves may be considered as planar in any small par of space [3]. While deriving he expressions of he Doppler effec and aberraion, Einsein emphasizes ha he source of elecromagneic waves is very far from he origin of 3
co-ordinaes ; and ha an observer is moving wih velociy v relaively o an infiniely disan source of ligh [4], obviously assuming ha a far disances from he source of wave (i.e. a he disances ha are much greaer han he lengh of wave) he differences beween spherical and plane waves shall become infiniesimal. Even if hose argumens were correc, ha would only mean ha he Lorenz ransformaion is an approximaion o he physical realiy, and a more precise heory would give correc resuls for any spaial dimensions. Below i is shown ha he expressions of he Doppler effec and aberraion for spherical and plane waves mainain heir differences even a infiniy. I is no surprising ha he applicaion of he Lorenz ransformaion for he spherical frons of elecromagneic waves gives disored picure of he realiy. This is, undoubedly, he weakes poin in he special relaiviy. The search for a beer ransformaion law is necessary. A correc ransformaion mus mainain he spherical form of elecromagneic waves in any inerial frame. Tha is he requiremen of he consancy of he ligh velociy in empiness and he isoropy of space. Le us ge he expression of he Doppler effec for spherical elecromagneic waves firs. I is obvious ha in any frame all waves emied by a source of an elecromagneic wave in empiness are conained beween he source and wave fron (a any insan of ime, he las wave, or a par of i jus emied, is a he source and he wave firs emied has jus arrived a he locaion of he wave fron). In any direcion he disance beween he source and he fron of he emied wave is equal o he sum of wavelenghs conained beween hem. Le in figure 1 he source of wave is fixed in he origin O 1 of he frame moving wih a consan velociy v=c. If he wave emied is a spherical one, hen all waves emied in ime in he saionary frame, n 0, (which is equal o he number of oscillaions wihin he source of wave) will be conained in he space beween he poin O 1 and sphere O. Assuming ha he ligh velociy is he same in boh frames, for some poin A of he wave fron in he saionary frame we have: cosθ + = n0λ0 1 θ + = 0λ θ O 1A = c 1 cos n, (3) where n 0 is he number of oscillaions wihin he source of wave in he inerval of ime in he frame in which he source of wave is a res; λ 0 is he lengh of he emied wave in he same frame; λ θ is he lengh of he wave emied in direcion AOO 1 = θ relaive o he direcion of he movemen of he source in he saionary frame. From (3): 0λ0 cos ' n0 n θ + λ θ =. (4) Someimes i is more convenien o use angle AO 1 C = α insead of angleθ. Angle α is he one a which, from he poin of view of he saionary frame, he ray OA (or he poin A of he wave fron) propagaes in he moving frame conneced wih he source. If we assume ha he ray OA propagaes hrough an imaginary moving 4
ube of a very small diameer, hen angle α is he slope of he ube as seen from he saionary frame. sinθ sinα = (5) cosθ + So, insead of (4) we can use he following equaion: n0λ 0 1 sin cos α α λ = α. n0 If he wave emied from he moving source in O 1 represens a hin ray of plane wave (i.e. he fron of wave lies in a plane perpendicular o he direcion of propagaion of he ray) propagaing in direcion OA, hen AK is he line of inersecion of he plane of he picure wih he plane of he wave fron a some insan of ime in poin O. In his case O 1K = c( cosθ ) = n0λ0 ( cosθ ) = n0 λθ, and n0λ0 ( cosθ ) λ θ =. (6) n0 From equaions (4) and (6) i is eviden ha in regard o Doppler effec spherical and plane waves mainain heir uniqueness a any disances. Hence, he adopion of he plane-wave approximaion for elecromagneic waves is no jusified and only leads o complicaions. We can consider he lenghs of elecromagneic waves as naural unis of measuremen of space inervals. We have good reason o assume ha any physical uni of measuremen of disances from a moving source o he fron of elecromagneic wave undergoes he same kind of changes as he lenghs of elecromagneic waves: U n0u 0 cosθ + θ = ; n0 U θ is a naural uni of lengh in direcion θ ; 0 hypoheical uni which undergoes no changes. When 0, hen U θ U0 i.e. naural unis coincide wih mahemaical unis when disances beween wo saionary poins or a saionary poin and wave fron are measured. Using angle α, we have: n0u 0 1 sin cos α α U = α. n0 In naural unis he disance beween he moving source O 1 and he fron of wave in any direcion is equal o: U n0 O 1 A = c 1 sin cos α α : α = c = O1 A1. U0 n0 Thus, when measuring disances from a moving source O 1 o he poins of he wave fron in naural unis, we ge a new sphere wih cenre in O 1 and radius U is a mahemaical uni, i.e. a 5
n0 r = R ; or vice versa, if we assume ha he fron of an elecromagneic wave in n empiness mainains is spherical form relaive o any poin of reference wheher saionary of moving, ha would mean ha any measuring device, including our eyes, uses naural unis of measuremen in he assessmen of disances from moving poins. n0 Now we have o prove ha =, as i is shown in figure 1: n 0 O1H = r = c. Indeed, i is obvious ha he spheres O and O 1 inersec and for he poins of inersecion n 0 λ α = n0 λ0, i.e. λ α = λ0. Also, O 1A = n 0 λα, O 1 B = n 0 λ π + α, and 1A O1B r O =, where r = O1 A1 = O1B 1 = n0 λ0 ; so, λα λπ + α = λ 0. Thus, for he poins of inersecion of he spheres: λα = λπ + α = λ0 ; hence α = ±π / and λ π / = λ0 From n 0 / = n0λ0 follows: n 0 = n0 and ±. λ π r = R. λ0 1 cosθ + Thus, λ θ = 1 The same expression was obained in [5]. I is obvious ha n 0 = /τ0 and n 0 = / τ 0 = / τ 0, where and are he readings of he clocks a he origins of he saionary and moving frames O and O 1 respecively, and τ 0 and τ 0 are he periods of oscillaion wihin he source of wave in is res frame and in he saionary frame respecively. Thus = and τ = τ / ; i.e. in a moving poin ime flows 0 0 1 1/ imes slower han in a saionary frame, or, wha is he same, any period of ime (e.g. he ime of a complee revoluion of a hand of a clock) is 1/ imes longer in a moving poin han in a saionary poin. I is ineresing ha in figure 1: A 1B1 = AB, sin α which coincides wih he equaion of he Fizgerald-Lorenz conracion, i.e. his expression is he same for spherical and planar elecromagneic waves. This simple geomerical fac, which is described in [6], poins also o he necessiy of inroducion of naural unis. From he poin of view of he saionary frame, he sphere of radius R wih cenre in saionary poin O is ransformed ino he sphere of radius r = R wih cenre in moving poin O 1. This means ha poins A and A 1 in figure 1 are he same poins considered relaive differen poins of reference. So, if in poin A of he saionary frame a clock is placed, which reads = τ, hen he same clock is placed in poin A 1 of he moving frame as well and is reading mus be =τ 1. Bu, as already menioned above, he same clock canno show differen imes. Tha leaves he 6
only possible soluion: he reading mus be zero (or in oher words, whaever he reading may be, we have o coun ime from ha poin). Thus, ime coordinaes of he poins of he fron of elecromagneic wave emied from he origin of he frame a =0 mus be zero. (If an elecromagneic wave is emied from he origin of he frame a = 0, hen he ime coordinaes of he poins of he wave fron is = 0 oo.) Then he readings of he clocks in he origins of he saionary and moving frames are = τ and = τ respecively. The clocks beyond hose spheres show negaive ime (his means ha he impulses ha arrive a hose poins have been issued earlier han he readings of he clocks a he origins of he frames). Le a = = 0 a moving clock O is in poin O of he saionary frame and an elecromagneic wave sars emiing from he same poin. Afer some ime he moving clock arrives a poin A and he wave fron arrives a poin C. In figure (a) we can see he picure in mahemaical unis: OA = c, OC = c, AC = O C = ( 1 )c. Bu he measuremen of he disance from he moving source o he fron of an elecromagneic wave mus give he resul in naural unis; oherwise he ligh velociy would no be independen from he velociy of he source. Indeed, from he OC O C poin of view of a saionary observer, if = c, hen c. 1 The naural uni in direcion O C is U = U0 ; so, in hose unis 1+ AC O C = = c 1. In his case (figure (b)) he poin of wave fron in U / U 0 direcion O C is spli ino wo poins: C and C (relaive o poins O and O respecively). This is he picure we should ge allowing for he fac of consancy of ligh velociy. In he realiy he poins C and C are he same poin and we ge he picure represened in figure (c), where A 1 A is equal o C C in figure (b), or he picure represened in figure (d), where O 1 O is equal o C C in figure (b). Disances OA and O O in figure (a) are incommensurable, because hey are o be expressed in differen unis. Tha means ha i is impossible for boh ends of hose segmens o coincide. In figure (c) he ends of hose segmens coincide in poin O, and in figure (d) he oher ends of hose segmens coincide in poin A. Figure (c) may be called he viewpoin of he observer a poin O and figure (d) he viewpoin of he observer a poin A. A saionary clock a poin A 1 in figure (c) (or a poin A in figure (d)) shall OA1 show 1 = = 1, which coincides exacly wih he reading of he moving c clock O. So, by comparing he readings of he clocks, i is impossible o ell which one of he poins A 1 and O (or A and O in figure (d)) is moving. 7
O C O A C (a) O C O A C (b) O C O A 1 A C (c) O C O O 1 A (A 1 ) C (d) Figure. An illusraion of he effec of naural unis. (a) All disances are in mahemaical unis; (b) disance O C is in naural unis; (c) he real picure for he observer in poin O; (d) he real picure for he observer in poin A. Thus, for he observer in poin O (figure (c)) he moving clock O is in poin A 1 and no in poin A; for he observer in poin A (figure (d)) he moving clock is in poin A, bu a he beginning he clock was in poin O1 and no in poin O. This may seem unbelievable and needing some addiional commenaries. When measuring he disance from some saionary poin o a moving poin by means of a ruler, we are subsiuing he disance beween hose wo poins by he disance beween wo saionary poins of he ruler. We canno be sure ha his is a legiimae operaion. The same is rue for he measuremens using radar: he ime he radar impulse requires for covering he way from he radar o a poin (saionary or moving) is equal o ha from he poin o radar, which is anamoun o measuring a disance beween wo saionary poins. Direc measuremen wih a ruler or radar gives no he real, bu he mahemaical value of he disance o a moving poin. If a direc measuremen of he disance o a moving poin gives OO = L = c, hen he real disance is: 8
L = c c When = 1, L = L. Thus, 1 1 = 1 L = 1 + L 1 L =, (7) c i.e. he difference in readings of wo synchronized clocks is equal o he ime required for ligh for covering he real disance beween hose clocks. I mus be noed ha here is nohing new in his simple mahemaical fac which follows from he special relaiviy as well. Eq. (7) is rue for any velociies (even for he zero one!), as well as for any pah of a moving poin. In he case of zero velociy, i.e. if boh clocks are saionary ones, L is equal o he real lengh of he pah which we consider a communicaing signal will make from one poin o anoher. Le O and A be wo marks on a ruler and OA = L. Then if in poin O in L figure (c) he clock shows, he synchronized clock in poin A shows A =. c Tha means: if a moving clock O arrives from poin O a poin A wih velociy v c, he difference in he readings of he clocks in O and O will be L = A = 1 1. If he speed of he moving clock v<c, he difference in c L L he readings of he same clocks will be: 1 = A = = <. Tha is 1 c c because in his case he disance covered by he moving clock is no OA = L, bu OA 1 = L 1 < L (we do no have o mix up he lengh of he ruler wih he disance covered by he clock). For he observer in poin A (figure (d)), he clock arriving a 1 A, as well as his own clock, shows some ime 1, and = ime ego he clock was a poin O 1 ; if v c, hen poin O 1 almos coincides wih poin O (he lef mark on he ruler); if v<c, hen he disance covered by clock is O 1 A = L 1 < L. Poin O 1 does no coincide wih he lef mark on he ruler. In figures 3(c) and 3(d), we can see ha he real disance covered by he clock is L 1 only, alhough, according o he ruler, he disance beween he firs and las poins of he clock s ravelling is L. This is an effec of he use of differen unis of measuremen, bu we can look a his in anoher way as well: insead of he movemen of he clock in he frame of he ruler, le us consider he movemen of boh of hem in some preferable frame, during which he clock O moving o he righ from he lef mark of he ruler, O, and he righ mark, A, of he ruler moving o he lef, arrive a poin A 1 simulaneously (figure (c)). In realiy no inroducion of a preferable frame is necessary, because his is only an illusion caused by he use of differen unis of lengh. The fac ha he movemen of he ruler is an illusion may be illusraed by he following example: le us imagine ha he earh does no roae around is axis and a je is flying wih velociy v along he earh s equaor. Since he speed of he je v<c,. 9
he real disance covered by i during is circumnavigaion will be less han he lengh of he equaor which in his case represens a ruler. This fac is due o he use of differen unis of measuremen for he disances o saionary and moving poins. We may explain his fac also by assuming ha he earh is roaing wih adequae velociy in opposie direcion in a preferable frame, bu his roaion will be undeecable for an observer on he earh s surface; i.e. his roaion is only an illusory one. To be more convincing in our reasoning le us use again he mehod of couning waves. The number of he waves emied from a saionary source of elecromagneic waves in poin O (figure (a)) in ime is n 0. If here is no moving poin O in A, hose n 0 waves are disribued beween he source in poin O and he poin of wave fron C in he following way: n0 waves are locaed beween O and A, and ( ) n 0 waves beween A and C. If now insead of saionary poin A we consider moving poin O, hen beween O and he poin of wave fron C here will be ' n0 = n0 waves. Indeed, from he poin of view of he saionary frame, ' ' AC = ( ) n 0 λ0 = n0λ0, from which n0 = n0. The remaining 1+ 1 1 n0 waves are locaed beween poins O and O. Since beween O and A, and beween O and O differen numbers of waves are conained, he disances OA and O O canno be equal o each oher. I is obvious ha he disances OA 1 and O O in figure (c) conain he same numbers of waves. Thus, he poin O is in poin A 1 of he saionary frame and no in poin A as we obain by means of measuremen in mahemaical unis. On he oher hand we canno exclude he possibiliy ha some kind of measuremen may give a real disance: OO r = c1( ); hen he mahemaical disance is OO m = c1, where 1 =. We have o expec ha direc measuremens wih a ruler or radar will always give mahemaical values, while some indirec measuremens (he ones which canno be reduced o measuremens wih a ruler or radar) may give real values. In D. Frisch and H. Smih experimen [7] here is no direc measuremen of he disances covered by muons. So, we canno be sure wheher he heigh of M. Washingon, H, is a mahemaical disance covered by muons or a real one. In he laer case, he disance covered by muons in mahemaical unis H would be H 1 =. The ime necessary for covering his disance by muons in H H 1 he saionary and heir res frames are = and = c 1 1 c 1 1 respecively. According o he experimenal daa H 1910 m, 0. 995, original 6 number of muons n 0 568, and heir half lifeime τ 0 1.5x10 sec. Those values yield = 0.708x10-6 sec. and he number of remaining muons, n 410. This value is 10
much closer o he experimenal value of 41 as agains he value of n 4 obained by he auhors in heir calculaions, bu his one paricular case, of course, does no prove anyhing. Only saisical analysis of series of similar experimens can give saisically significan answer wheher our suggesion is rue or no. Now le us find ou wha he fron of he elecromagneic wave is like in a moving frame. Le in Fig. 1 he number of waves emied from he origin of he saionary frame, O, during ime period of is n 0, hen OA = n 0 λ0. If his array of n 0 waves is wached from he moving frame, hen for a spherical wave poin A becomes some poin A on he line OA: O1 A c cosθ + O 1A = n0λ θ = n0 =. n 0 The se of poins A for all possible direcions forms he surface of a sphere c c wih cenre in O where O O1 =, and radius R = O A =. Thus, we have go wha we were looking for: he wave fron in he moving frame a he insan of ime when he clock in poin O shows ime. In oher words his means ha in moving frames, while describing he propagaion of spherical elecromagneic waves, he unis of ime and space inervals are conraced in 1/ imes and his conracion occurs uniformly in all direcions, which are radically differen from he predicions of he Lorenz ransformaion. Now we can wrie new expressions for he ransformaion of he frons of spherical elecromagneic waves. If he origin of he moving frame is in poin O 1, hen: x c y z x =, y =, z =, = = 0 ; and are ime coordinaes of he poins of he wave fron. If we assume ha he origin of he moving frame is in poin O, hen: x y z x 1 =, y =, z =, = = 0 ; (8) We have o add here he following relaions as well: o O = 1 1 O, O =, (9) were O, O and 1 O are he readings of he clocks in poins O, O 1 and O (he cenres of he spheres in Fig. 1) respecively. Equaions (8) and (9) are he counerpars of equaions (1) for spherical elecromagneic wave. In he case of a plane wave: O1F c( cosθ ) n 0λ θ = n0 = = O1 A. n 0 11
To find he locaion of poin A, le us noice ha in direcion X (he direcion of movemen of he frame) i is impossible o make disincion beween he spherical and plane waves. Thus, X componens of hose waves mus be he same in any frame, i.e. x coordinae of poin A mus be he same as ha of poin A. So poin A lies on sinθ he segmen A G (figure 1); hen A G = AF and, from OA O1, sinϕ =, cosθ where ϕ = A O 1 G ( ( ϕ θ ) is he angle of aberraion for plane elecromagneic waves), as follows from he Lorenz ransformaion. In he case of spherical waves angles ϕ and α coincide. Thus, according o new ransformaion law he slope of he above-menioned ube is he same in he saionary and moving frames and he angle of aberraion is ( α θ ). The relaion beween hose angles is given by (5). In figure 1 we can see ha using plane wave approximaion insead of spherical wave approximaion for elecromagneic waves, we are subsiuing poin A by poin K. Tha is he size of he error we make using he Lorenz ransformaion. Le hree clocks clock 1, clock, and clock 3 be placed in O, O and A poins respecively. A he beginning, when all of hem show zero ime hey coincide in poin O of he saionary frame. Clocks and 3 are moving relaive saionary clock 1 wih velociies v1 = 1c and v = c respecively, θ is he angle beween hose velociies. Fig. 3 represens he picure afer some ime 1 by he saionary clock 1. OO = 1 c 1 and OA c1 3 = 1. show = ; clock shows In he res frame of clock, when clock 3 shows and clock 3 = 1 1 3 = 1, clock mus = 1, (10) 3 because, as we have seen, i is necessary ha = ( ) +. In his frame he disance beween clocks and 3 is no O A, bu O 1 A, where ' O A O = = 1A c. (11) 1 1 Tha is because in he res frame of clock he uni of lengh is1/ 1 imes shorer han in he res frame of clock 1, and any disance, in paricular he disance beween clocks and 3, mus be he same imes larger. 1 1
A O θ O O 1 Figure 3. An illusraion of he law of addiion of velociies. In Fig. 3 ( + cosθ ) c O A =. (1) 1 1 1 From (10), (11) and (1): + 1 1 cosθ =. 1 1 cosθ + 1 Le in Fig. 1 a Michelson-Morley device is oriened in direcion α relaive o axis X along which he earh is moving. The lengh of each arm of he device (a oneway disance covered by he ligh in each direcion in he res frame of he devise) is R. From he poin of view of a saionary frame in which he device (ogeher wih he earh) is moving wih velociy along he axis X one arm of he device is O 1 A and he oher - O 1 B. In naural unis hose arms (O 1 A 1 and O 1 B 1 respecively) are equal and he ligh emied from poin O a = = 0 arrives a boh ends of he device simulaneously no only in he saionary, bu also in he moving frame, afer he periods of ime = τ and =τ 1 respecively. Thus, in he Michelson-Morley experimen no only he oal imes of he round rips of ligh impulses in opposie direcions are equal in any frame, bu also he imes of he one-way back and forh ravels of hose impulses. 13
Now le us have a brief overview. The use of he Lorenz ransformaion, which assumes he planar characer of elecromagneic waves, in he cases of he frons of spherical elecromagneic waves, as i was o be expeced, leads o a disored picure of physical realiy. The new ransformaion developed in his paper mainains spherical form of elecromagneic waves in all inerial frames. According o his ransformaion, conrary o he predicions of he Lorenz ransformaion, he unis of ime and space inervals in moving frames are conraced in 1/ imes and his conracion occurs uniformly in all direcions. I seems eviden ha such ransformaion won have any effec on he mahemaical expressions of physical laws. The expressions of Doppler effec and aberraion for spherical elecromagneic waves derived here differ from he similar expressions for planar elecromagneic waves and hose differences persis up o infiniy. Proceeding from he fac of consancy of ligh velociy in empiness and isoropy of space, we have o assume ha he unis of lengh undergo he same kind of changes as he lenghs of elecromagneic waves, and naural unis of lengh are o be inroduced. Naural unis coincide wih mahemaical ones when he disances beween wo saionary poins or beween a saionary poin and he wave fron are measured. Using naural unis we ge ha he imes of back and forh ravels of an elecromagneic impulse during is round rip mus be equal in any frame. In order o avoid logical difficulies i is necessary o assign o he poins of he fron of an elecromagneic wave, emied from he origin of a frame a =0, zero ime coordinaes; oherwise he arrival of an elecromagneic impulse a some poin of a saionary frame a some insan of ime in ha poin, which mus be an invarian, will occur a differen imes in differen frames (his simple fac has been srangely overlooked by he physiciss for so many years). In order o mainain he correc balance of he numbers of emied elecromagneic waves he real disances covered by moving poins are o be inroduced. A real disance beween a saionary and a moving poin conains he number of waves which is equal o he difference in he numbers of waves conained beween he fron of he wave and he saionary and moving poins respecively. In oher words, he difference beween he readings of he synchronized saionary and moving clocks is equal o he ime necessary for he ligh for covering he real disance beween hose clocks. A correcion o he expression of he law of addiion of velociies is made. I is encouraging ha he new approach is much simpler and seems o be a beer approximaion o he realiy. A horough experimenal check-up of all heoreical resuls of he relaivisic heory is absoluely necessary. References: [1] Pierce, J., Almos All Abou Waves (The MIT Press, Cambridge, Massachuses and London, 1974). [] Feynman, R., Leighon, R., Sands, M. The Feynman Lecures on Physics, Vol.1 (Addison-Wesley, Reading, Massachuses, Palo Alo, London, 1963). [3] Landau, L., Lipshis, E. Theoreical Physics, Vol., Field Theory (Nauka, Moscow, 1967). 14
[4] Einsein A. Zur Elekrodynamik beweger Körper, Annalen der Physik 17, 891-91. [5] Bregadze T. The Symmery of Measuremen, CERN: EXT-004-140. [6] Bregadze T. A Propery of Sphere and Relaiviy, CEERN: EXT-004-038. [7] Frisch, D. and Smih, J. "Measuremen of he Relaivisic Time Dilaion Using Mesons", Am. J. Phys. 31 (1963) 34. 15