Lorentz Transform in Multi-Dimensional Space

Transcription

1 Journal of Modrn Physics Publishd Onlin Novmbr 01 ( Lorntz Transform in Multi-Dimnsional Spac I. A. Urusovskii Acoustics N. N. Andrv Institut Moscow Russia urusovskii_ia@mail.ru Rcivd August 3 01; rvisd Sptmbr 30 01; accptd Octobr 9 01 ABSTRACT It is shown that in Euclidan spac with any numbr of spatial dimnsions mor than thr th Lorntz transform holds tru if th propr tim of ach lmntary particl is proportional to th lngth of its path in th tra-dimnsional subspac and all lmntary particls mov at th spd of light in th complt spac. Th si-dimnsional tratmnt of th Coulomb forc of intraction btwn two chargs is givn. Th lctric forc is du to th motion of chargs in th tra-dimnsional subspac and is qual to th corrsponding Lorntz forc. Kywords: Lorntz Transform; Euclidan Multi-Dimnsional Spac; Compton Wav Lngth; D Brogli Wavs; Spin and Isotopic Spin; Hisnbrg Uncrtainty; CPT Symmtry; Coulomb Forc 1. Introduction It is commonly assumd that spac-tim is psudo- Euclidian whil th intrval ds of rlativity thory is an lmnt of th trajctory of a particl in this spac-tim. But for any Euclidan spac having mor than thr spatial dimnsions th Lorntz transform may b obtaind if th lmntary particl is moving at th spd of light in th complt multi-dimnsional spac and th propr tim of this particl is proportional to its path lngth in tra-dimnsional subspac (Y) of th whol spac. This plains th twins parado as wll: th mor rapidly a body movs in thr-dimnsional subspac (X) th lowr its vlocity in tra-dimnsional subspac and consquntly th slowr its propr tim passs. This assumption gos back to F. Klin s ida [1] of th motion of particls with th spd of light in multi-dimnsional spac whr mchanics is rprsntd as quasi-optics. Th light and also particls of substanc hav corpuscular as wll as wav proprtis ampls of which ar th diffraction of lctrons whn thy ar rprsntd as a wav and photolctric mission whn photons ar rprsntd as particls. Ths proprtis tstify that svral basic proprtis of light and particls ar th sam. Th main proprty of light is that it propagats in th absnc of gravitation at th spd of light in any rfrnc fram. Thn lmntary particls of substanc must also mov at th sam spd. This is impossibl in thr-dimnsional spac but possibl in multi-dimnsional spac. All dirctions in tra-dimnsional subspac Y of th multi-dimnsional Euclidan spac considrd ar prpndicular to any dirction in thr-dimnsional subspac X. Thrfor projctions ds and d of a displacmnt cdt of a particl moving at th spd of light in th complt Euclidan spac of any numbr of spatial dimnsions mor than thr on subspac X and subspac Y rspctivly ar connctd by th Pythagoran thorm: cdt ds d whr cdt is a displacmnt of th particl par tim laps cdt. From this on obtains th mtric ds cdt d. (1) Particls should b confind to a small vicinity of th thr-dimnsional Univrs by th forcs (of cosmological natur) orthogonal to it. In th Euclidan spac undr considration lmntary particls cannot b wll off th subspac X at larg Compton distancs and must b confind nar this subspac. Othrwis thr would b nithr compact trajctoris of lmntary particls in tra-dimnsional subspac nor macroscopic bodis. Such forcs (of th Lorntz typ) ar prpndicular to any dirction in thr-dimnsional subspac and ar th rason why th trajctoris of lmntary particls ar compactd in tra-dimnsional subspac dspit th cntriptal forcs in it. Thus projctions of ths trajctoris on additional subspac ar finit ons and lmntary particls mov at th spd of light rmaining at rst on avrag in this subspac. For this rason atoms and macroscopic bodis consisting of lmntary particls may b at rst as a whol. Th disprsion quation is th sam for th acoustic wavguid th lctromagntic on and d Brogli wavs: v v ph g c whr v ph is th phas vlocity v th group vlocity c th spd of wavs in a fr g Copyright 01 SciRs.

2 1750 I. A. URUSOVSKII mdium (th spd of sound in th first cas and that of light in th othr two cass). Th main charactristic of any wavguid is that it has finit transvrs dimnsions. Th disprsion of wavs is du to ths dimnsions only. This indicats that th part of spac which w dal with in th primnt is only approimatly thr-dimnsional and has rathr small (Compton) sizs in th additional subspac. Enrgy propagats along rays. That is why th spd of its propagation along th wavguid is lss than th spd of light and is qual to csin th group vlocity whr is th angl of inclination of th trajctory of an lmntary particl to th transvrs sction of th wavguid. Wav fronts ar prpndicular to th trajctory. Thrfor ths fronts mov along th wavguid mor rapidly than th spd of light and thir phas vlocity is c sin th spd of an imagind tachyon so that th product of th group and phas vlocitis is c. Th disprsion is formd by th walls of th wavguid if it has thm or forc filds as for an lctron moving in a magntic fild along a hlical lin; this is also a wavguid of a sort. Hr th disprsion quation is du to th prpndicularity of trajctoris to th wav fronts. Th nrgy of a photon is qual to h whr is th frquncy of light and h th Plank constant. By virtu of a principl of similarity of th basic proprtis of substanc and light concrtizing th principl of simplicity [3] th rst nrgy mc of a particl may b rprsntd as a quantum of nrgy h so that mc h () Th uniqu and natural frquncy for a particl of substanc at rst in X is th frquncy of its rotations in tra-dimnsional subspac Y. On th othr hand th particl movs with th spd of light along th dirctri of th motion tub from which πa c whr a is th radius of th tub. Eliminating from this quality and () on finds πa h mc a mc; that is th lngth of th dirctri is qual to th Compton wavlngth. Th lngth of th dirctri corrsponds to th priod h of th coordinat of action in 5-optics [4]. Thr is no nd to considr th complt spac as a psudo-euclidan on. Lt us dmonstrat that th Lorntz transform may by obtaind for th Euclidan spac undr considration. Th particl movs along th cylindr surfac (th motion tub as th nvlop of th godsics) with Compton radius a mc and with its ais in th subspac X and dirctri in th subspac Y. Th intrval ds is a projction of th displacmnt of a particl on th dirctri of th motion tub and d is a projction of th displacmnt on th tub ais (s Figurs 1 and ). Th particl which is stationary in th projction onto X in th inrtial fram of rfrnc K movs at th spd of light c and travls in th simplst cas in a circl within th thr-dimnsional subspac Y that complmnts X with rspct to th complt Euclidan spac R6 whil th cntr of th circl is in X. In any othr inrtial fram of rfrnc this particl movs along a hlical lin (curv 1 in Figur ) locatd on th cylindrical surfac (motion tub) in R with its ais blonging to X. 6 Lt us assum that th propr tim of th particl is proportional to th lngth of its path in Y bcaus it is a natural masur of tim. This lngth is proportional to cos whr is th angl of inclination of th hli with rspct to th tub dirctri. If th particl maks on rotation within th propr tim for a fid obsrvr with rspct to whom th particl movs along th tub with th vlocity v csin this will occur within th tim v csin whr cdt d Figur 1. Projctions ds and d of a displacmnt cdt of a particl moving at th spd of light in th complt multi-dimnsional Euclidan spac. A s O ς Figur. 1 is th hlical trajctory of a particl moving in si-dimnsional spac with th spd of light c along th cylindr surfac (th nvlop of th godsics) of Compton radius a= mc with its ais in th subspac X and dirctri in th subspac Y. ς =ct. is a hlical lin of qual propr tim of this particl. It passs through th particl prpndicularly to th hlical trajctory. It movs along th sam cylindr surfac with th vlocity of th d Brogli wav. Its pitch is qual to th d Brogli wavlngth. In subspac X th particl movs at th spd v=csin whr is th angl of inclination of th trajctory of an lmntary particl to th transvrs sction of that cylindr. B 1 P y y 3 ds Copyright 01 SciRs.

3 I. A. URUSOVSKII 1751 sin vccos 1 vc. (3) In (3) and th subsqunt discussion th plus sign applis to th particl rotating about th ais of th tub in th positiv dirction and th minus sign applis to th antiparticl rotating in th opposit dirction. Intrvals of th propr tim of th particl (or antiparticl) d and of th fid obsrvr s tim dt ar rlatd by th formula dt d cos d 1 v c. (4) In th fid fram of rfrnc K th particl has a componnt of th vlocity ccos along th dirctri. For th fid obsrvr th propr tim of th particl is according to (4) also proportional to cos so that th particl movs with th vlocity c in its propr fram K of rfrnc as wll. A particl at rst in K a particl moving with th spd of light c along th dirctri displacs pr propr tim d in an intrval ds qual to ds cd. (5) Th momntum of this particl is a vctor dirctd along th tangnt to th dirctri at a point whr this particl is placd at a givn tim. Th magnitud of this vctor is mc which is th product of th mass m of th particl and its spd c. This is th momntum at rst in rlativistic mchanics. Th nrgy at rst E according 0 to this dfinition is th product of momntum and th spd of a particl: E0 mc. In th gnral cas th total momntum of a particl is th vctor dirctd along th tangnt to th hlical trajctory. Its valu p is th product of th mass m of a particl and th ratio of its path d cdt (6) in th whol spac to th propr tim d pndd for this path: d mc p m mc 1v c d cos. This is th rlativistic formula for th total momntum of a particl. Projctions p and py of th total momntum on th gnratri and dirctri of a tub ar qual to th spatial and tmporal componnts of th four-momntum of a particl rspctivly [56]: y p mctan mv 1 v c p mc. (7) In th gnral cas 0 and th total nrgy of a particl E is th product of th total momntum p and th spd of movmnt c along a hli: mc E pc mc cos 1 v c. (8) This valu is th total rlativistic nrgy of a particl. Not that th ratio of th total nrgy to th total momntum of th particl is th sam as that for a photon. This is yt anothr common proprty of light and sub- stanc. Displacmnt of th particl ovr th intrval ds along th motion tub dirctri and th corrsponding rotation through th angl d ds a about th ais of th tub whr a is th radius of th tub ar idntical in any fram of rfrnc. Dsignating th projction of particl displacmnt d ovr th surfac of th tub onto its gnratri by d in th fram of rfrnc K and using th Pythagoran thorm for th rctangular triangl OAB shown in Figur on obtains (1). If this rlation is considrd as th initial on thn (6) follows from it; that is th particl movs in R6 at th spd c. Th projction of th sids of th triangl OAB on th trajctory of th particl givs scos sin. (9) Lt us writ th initial conditions in th form t 0 with s 0. Thn rfrring to (4) and (5) it follows that: s c ct. (10) Substituting sin vc () and (10) into (9) givs th Lorntz transform for tim: sin cos 1 t c t v c v c A similar considration applid to th systm of rfrnc K taking into account that th systm K movs rlativ to th considrd particl with vlocity v lads to th rvrsd transform: t c sin cos vc vc 1. whr is th coordinat along th lin of qual propr tim. Th transition from th systm K to K corrsponds to a turn through an angl about th origin s 0 of coordinat nt s on th surfac of th motion tub togthr with trajctoris of particls on it. This turn transfrs a hlical trajctory in th dirctri of th tub. For a gomtrical intrprtation of th rst Lorntz transform lt us considr a trajctory of a particl moving along th tub with th sam vlocity v and intrscting at a tim t 0 with th hli scos sin 0 at an arbitrary point P. In th systm of rfrnc K trajctoris inclind at th angl to th dirctri ar th lins of th constant coordinat of th systm K. Th coordinat BP is masurd along th hli dscribd by Equation (9). Th masurmnt is takn from th normal sction of th tub vt sin to. Copyright 01 SciRs.

4 175 I. A. URUSOVSKII th sction rachd by th particl P at tim t whn is th distanc btwn P and th dirctri OA. Projcting sgmnts and s (s Figur 3 as th rsolvnt of th cylindr in Figur ) on th gnratri and dirctri th trajctory of th particl and th hli (along ) prpndicular to th trajctory ar obtaind with cos > 0: cos sin scos sin cos sin s cos ssin. Dividing ths qualitis through by cos and liminating s and by mans of (3) and (10) according to which s c ct sin vc and cos 1 vc in th considrd cas on may asily obtain th Lorntz transform in th standard form: 1 vt v c v c 1 v c t t v c 1v c 1 vt v c. Th propr lngth of a moving rigid scal is th diffrnc btwn th coordinats of its nds. In th systm K this is qual to th lngth of a sgmnt of th hli prpndicular to th trajctoris of particls moving with this sgmnt btwn normal sctions of th motion tub corrsponding to thos nds. It is a sgmnt of th lin of qual tim in th systm K. Th lngth of th sam scal in th systm at rst K is th diffrnc btwn th coordinats of its nds. This is qual to thdistanc along th gnratri btwn thos normal A O ς B Figur 3. Th volvnt of th motion tub shown in Figur ; s and ar coordinats of a particl along th dirctri of th motion tub and th tub ais rspctivly in th systm K at rst; is th sam but along th gnratri in th systm K connctd with th particl. ς =ct OA = PD = s AP = OD = BP = OC = CP = OB = ς. C ς D P sctions which is 1cos lss than th propr lngth. Thus th Lorntz contraction of moving scals is a rsult of th projction of lngths in multi-dimnsional spac on thr-dimnsional spac. Non-simultanity of spatially spacd vnts in on systm of rfrnc with simultanity in anothr is plaind by non-paralllism of hlis of qual tim in systms of rfrnc moving rlativ to on anothr. Th abov intrprtation of Formula (4) also holds for th curvd ais of a motion tub bcaus in any cas all normal sctions of such a tub ar prpndicular to any dirctions in th subspac X to which th ais of th tub blongs. Anothr hli placd on th sam tub prpndicularly to th hlical trajctory of a particl and passing through th particl is th lin of qual propr tim of th systm K. This hli movs along th tub with th vlocity of th d Brogli wav vph c sin c v whr v is th vlocity of th particl in th subspac X. Th pitch of this hli is qual to th d Brogli wavlngth a h h π h p 1 v c tan mc tan mv as is sn from (7) and Figurs and 3. Th angl coordinat s a of th hli dscribd by (9) and (10) is qual to s ct sin mc tan h π From a acos a cos this and from (6) and (7) it is sn that s a is qual to th phas of th d Brogli wav Et p. In th plac of th position of th particl = vt this phas is an angl of rotation of th particl on th motion tub. Th function pis a satisfis th Klin-Gordon quation a a. Th Hisnbrg uncrtainty rlations ar du to uncrtainty of coordinats and momnts of a particl in Y. In fact lt th dirctri of a motion tub of a particl b displacd in th plan y y3. Thn projctions of th momntum of a particl on as y and y3 and coordinats of th particl along ths as ar qual to py mcsin py3 mccos y cos y3 sin mc mc whr is th angl of rotation of th particl about th ais of a tub rckond from th ais y. Th avrag ovr valus of coordinats and projctions of th momntum ar qual to zro but thir man-squar valus ar qual to 1 1 y y3 py py3 mc mc Copyright 01 SciRs.

5 I. A. URUSOVSKII 1753 from which th following rlations can b found: py y py3 y3 4. Th simpl gomtrical tratmnt of spin and isotopic spin nds thr additional spatial dimnsions. In sidimnsional Euclidan spac spin and isotopic spin ar tratd as projctions of th total momntum on th subspac X and subspac Y rspctivly whil th propr magntic momnt is a rsult from rotation of th charg at th spd of light in Y along th orbit of th Compton radius [5-7]. Th first substantiation of spac si-dimnsionality was givn by di Bartini [8] who calculatd th fundamntal physical constants. A four-dimnsional tratmnt of Mtric (1) with a somwhat diffrnt account of wav proprtis of spac (as applid to four spatial dimnsions) is givn by Gribov [9]. Th propr momnt of momntum S of a particl is a vctor product of th propr momntum and radius vctor of this particl. Th componnt of th radius vctor and th componnt of vlocity of th particl on th ais of th motion tub ar prpndicular to th plan of rotation in Y and thrfor do not mak any contribution to S. Hnc for a particl moving in si-dimnsional spac along a hli but consquntly in a straight lin in a projction on X S is a vctor product of th projction of momntum and th radius vctor of this particl on Y. In this cas th magnitud of momntum S bcoms S S pya mc mc. This formula rtains som arbitrarinss in th orintation of vctor S in si-dimnsional spac: it may b orintd in any dirction in four-dimnsional subspac prpndicular to th plan of rotation in Y. In th gnral cas vctor S has four non-zro componnts along dirctions prpndicular to ach othr and th plan of rotation of th particl in Y. In th cas of rotation in th plan y y3 such componnts ar S1 S S3 S4 along th as y1 rspctivly and S S1 S S3 S4. Componnts S 1 S and S3 ar componnts of th spin of th particl and S 4 is a projction of th isotopic spin of th particl. Thus spin and isotopic spin ar th projctions on X and Y rspctivly. From (6) p y is indpndnt of vlocity v. Hnc spin and isotopic spin ar also indpndnt of vlocity v and ar not subjct to th Lorntz transformation. Vctor S which rmains prpndicular to th plan of rotation of th particl has thr dgrs of frdom and may b orintd in an arbitrary mannr rlativ to thos coordinat as. A uniform distribution of componnts of th vctor ovr th abov four as which ar prpndicular to ach othr and th plan of rotation in Y corrsponds to particls with spin on-half. Thn ths componnts ar qual to or and th sum of squars of ths componnts in X is qual to 34. In quantum mchanics this is th total (in thr-dimnsional spac) squar of th propr momntum of a particl. Orintations of vctor S obtaind from prvious orintations through allowabl turns kping on or two givn componnts invariabl ar rfrrd to th last cas as wll. So if on of th componnts of th vctor in X and on componnt in Y hav a fid valu or thn th vctor rtains th possibility of rotating about two corrsponding as. In this cas two nonfid componnts will not hav spcific valus (ths ar ordinary situations in quantum mchanics whr th absnc of fiation of quantitis is th cption rathr than th rul). For qual allowd probabilitis of orintations of that vctor th man-squar componnts mntiond abov ar qual to. A chang of dirction of rotation of a particl about th ais of th motion tub in th opposit sns also changs th signs of th componnts to th opposit ons and corrsponds to th transition to an antiparticl. In th gnral cas th momnt of momntum has four non-zro componnts along dirctions prpndicular ach to othr and a plan of rotation of a particl. Thrfor th thory of spin and isotopic spin must plicitly or implicitly us four coordinats and four projctions of vctors on th as of thos coordinats. Th total momnt of momntum M in R6 is th vctor product of th total momntum p mc and radius vctor r + a of a particl in R whr 6 p and r dnot th momntum and radius vctor in X and mc and a dnot th momntum and radius vctor in Y. Momnt M is a four-dimnsional vctor prpndicular to th plan of rotation of a particl in Y. On avrag ovr a priod of rotation about th ais of th tub th cross-trms disappar and thn M = L + S whr L is th orbital momnt in X and s a mc is th spin-isotopic spin momnt of rotation in Y. Thr componnts of S rprsnt th spin projctions S 1 S and S 3 on X and th componnt on Y rprsnts th isotopic spin S 4. Hnc on account of th mutual prpndicularity of vctors a and c and qualitis a a c c on obtains S. With th uniform distribution of componnts on four coordinat as which ar prpndicular to th plan of rotation in Y on finds Sj j S1 S S Of intrst is th qustion of why th valus of th propr momntum and its componnts in X and Y that is spin and isotopic spin ar indpndnt of th mass of an lmntary particl. In si-dimnsional tratmnt th answr is obvious: th momntum is proportional to this mass but th radius of th Compton orbit in Y for this particl is invrsly proportional to this mass and thrfor th product of th momntum and radius of th Compton orbit is indpndnt of this mass. Th propr magntic momnt of a chargd lmntary particl is dfind similarly to th propr mo- Copyright 01 SciRs.

6 1754 I. A. URUSOVSKII mnt of momntum S accordingly to th known formula of lctrodynamics [10]: c Rc whr R is th si-dimnsional radius vctor of th particl and c is th vctor of its vlocity in Y. Sinc a contribution to this vctor product givs only th projction a of th radius vctor R on subspac Y on finds ac. From this accounting for th mutual pr- c pndicularity of vctors a and c as wll as qualitis a a and c c on finds th magnitud of th propr momnt of th particl which is qual to th Bohr magnton: a B. (11) mc In th simplst cas whn th vctor has no componnts in subspac Y th componnts of this vctor dfin in X a thr-dimnsional vctor whos magnitud is qual to th Bohr magnton. A projction of th magntic momnt onto an arbitrary chosn dirction (calld th ais of quantization) in subspac X may hav a fid valu only in th cas whn th projction of th propr momnt of momntum has a fid valu as wll. In this cas according to (11) B. With a uniform distribution of componnts of th propr momnt of momntum ovr four as which ar prpndicular ach to othr and a plan of rotation of a particl in Y in th cas considrd S mca which quals 1 or 1 (in units of ). Thrfor S mc in accordanc with th primnt of Strn and Grlach. A particl which is at rst in X travls in th circl of radius a at th spd of light c. Th abov cntriptal cosmological forc in Y corrsponding to this circular motion is F0 pc y a mc a c ag tims largr than th wight of th particl at th Earth s surfac. It is for an lctron. Th sam rsult is obtaind whn th particl movs along th hli: F0 pck cos whr K cos a is th curvatur of th hli. Hr it is vidnt that F0 mc a at any. Lt us show that th forc of lctrical intraction btwn two lctrons in si-dimnsional spac is R th gnralizd Coulomb forc if th Biot-Savart formula holds tru for this spac whr R is th distanc btwn chargs in th complt spac. Th disposition of two lctrons on th opposit sids of th sam tub of motion has an nrgtic advantag whrby th distanc btwn thm in th complt spac is qual to R r 4a whr r is th distanc btwn projctions of th particls onto X and a is th distanc of thm from th ais of thir rotation in Y. Th tub radius a hr dpnds on r and tnds asymptotically to a mc with incrass in r with m and c bing th mass of a particl and th spd of light at infinity rspctivly. Du to such a rotation with th shift in phas π btwn two particls th Coulomb forc of thir rpulsion in th complt spac is qual to R whr is th charg of th lctron. Projctions of this forc onto subspacs X and Y ar F R sin and F R cos rspctivly whr 3 sin rr and cos ar so that F r R 3 F ar and F F R. Th forc F racts against th cntriptal forc F0 mc a. Thrfor th radius of rotation a is a littl in css of th tub radius a at infinity. Applying th Biot-Savart formula to si-dimnsional spac th total magntic fild of th charg at rst in X is dfind at th distanc R from th charg as Htot c R0 cr whr R0 is th unit vctor dirctd from th charg to th point of obsrvation and c is th vlocity of th charg. Whn R is th distanc btwn two lctrons R0 r0sin a0cos r rra ar 0 0 Htot c0 R 0 cr (1) c0 r0rrc0 a0ar R whr r0 is a unit vctor along th radius vctor r in X a0 is th unit vctor along th radius vctor of th charg in th plan of rotation in Y and c0 is th unit vctor along vlocity c. Lt us show that th Coulomb forc of intraction btwn th two chargs ( and ) is th Lorntz forc acting on ths chargs moving in Y. Rfrring to (1) this forc is qual to f c Htot c 0 0rR+ 0 0aR cr c c r c c a From this and taking into account that for two intracting lctrons c c f 0 0 0rR aR R c c r c c a Rvaling th tripl vctor products and taking into account th mutual prpndicularity of th vctors involvd and th fact that in th cas undr considration c c on obtains Copyright 01 SciRs.

7 I. A. URUSOVSKII 1755 a c0 c0 r0 r0 c0 c0 0 a0 f rr. 3 0 aa 3 0 R R In th last formula th first trm rprsnts th projction of th gnralizd Coulomb forc onto X and th scond trm is its projction onto Y. Thir magnituds ar qual to F and F rspctivly. From this it is sn that th lctric forcs (of th Lorntz typ as wll) ar du to th movmnt of chargs in subspac Y in th magntic fild arising bcaus thy ar orbiting in Y in contrast to th usual magntic forcs which ar causd by th movmnt of chargs in th sam subspac X. Th forc F is qual to zro at r 0. This is th point of indiffrnt quilibrium nar which lctrons may b comparativly slow moving for a long tim if thy ar jctd for had-on collision on to othr with th appropriat original kintic nrgy [7]. Whn thr is a chang of th cours of tim th dirction of rotation of particls in Y is rvrsd which causs th signs of th filds to chang to th opposit ons. In this way th corrsponding trajctoris in th complt spac occur as though thy wr rflctd from a mirror. Th motion of a particl along th hli (of th Compton radius in Y) with a rvolution to th lft (right) viwd in th dirction of travl is changd into motion along th mirror-rflctd hli with rotation to th right (lft). Th sign of th charg may b rgardd as nothing but a mark corrsponding to on or th othr (positiv or ngativ) dirction of rotation of a particl in th spac of tra dimnsions. In contrast to th standard formulation of th CPT thorm in which th proprtis of particls and antiparticls rspctivly undr th dirct and rvrs courss of tim ar jutaposd in th si-dimnsional tratmnt of CPT symmtry th proprtis of th sam lmntary particl ar jutaposd undr th dirct and th rvrs cours of tim. In this tratmnt th chargs of particls and antiparticls ar th sam but th signs of th corrsponding lctrical and magntic filds ar dfind by th dirction of rotation in th tra-dimnsional spac. Th corrsponding formulation of th thorm is as follows. If th cours of tim is rvrsd th particl movs in th complt spac backward along th sam trajctory of this particl as undr th dirct cours of tim. In this way th signs of th filds automatically chang to th opposit ons and th trajctory viwd in th dirction of travl in th complt spac occurs as though it wr rflctd in a mirror so that this particl acquirs all of th proprtis of th antiparticl [7]. Inrtia and th prssur of light ar phnomna of th sam natur bcaus all lmntary particls mov in th complt spac at th spd of light. Both ar du to changs of orintation of th momntum of lmntary particls in th complt spac undr th action of an trnal forc. Th si-dimnsional tratmnt of gravitation which corrsponds to th motion of particls with th spd of light in th Compton nighbourhood of th thr-dimnsional spac along th godsics complying with th Frmat principl lads to th Papaptrou mtric and gravitational wavs [11-13]. Th nvlop of th godsics has th form of a tubular surfac with th Compton transvrs sizs in th additional subspac whr th radius and spd of light vary along th tub. Th forc of gravity is th projction of cosmological forc F 0 on th mridian of that tubular surfac. Gravitational wavs which ar prturbations of ths radii and spd turn out to attnuat ponntially hr. Thir amplituds ar considrd in th nar-fild zon of th rotator with n Malts cross lobs and ar calculatd at n = 4 [1]. REFERENCES [1] F. Klin Ubr Nur Englisch Arbitn zur Gsammlt Matmatish Abhandlungn Springr Brlin 19. [] F. Klin Vorlzungn übr di höhr Gomtri Ohn Auftrag in Brlin Vol p. 19. [3] A. A. Margolin Principl of Simplicity Chmistry and Lif Vol p. 79. [4] Yu. B. Rumr Invstigations on 5-Optics Gostkhizdat Moscow [5] I. A. Urusovskii Si-Dimnsional Tratmnt of th Rlativistic Mchanics and Spin Mtric Gravitational Thory and th Epanding Univrs Uspkhi Sovrmnnoi Radiolctroniki. Zarubzhnaya Radiolctronika Vol p. 3. [6] I. A. Urusovskii Si-Dimnsional Tratmnt of th Quark Modl of Nuclons Uspkhi Sovrmnnoi Radiolctroniki. Zarubzhnaya Radiolctronika Vol pp [7] I. A. Urusovskii Si-Dimnsional Tratmnt of CPT- Symmtry Procdings of Intrnational Scintific Mting Physical Intrprtations of Rlativity Thory Moscow 4-7 July 005 pp [8] R. O. di Bartini Svral Rlations btwn Physical Constants Doklady Akadmii Nauk SSSR Vol. 163 No pp [9] I. A. Gribov Dark Mattr as Pico-Windows to Physically Equal Multivrs /13_1/01_gribov.pdf [10] J. D. Jackson Classical Elctrodynamics Wily Nw York & London 196. [11] I. A. Urusovskii Gravity as a Projction of th Cosmological Forc Procdings of Intrnational Scintific Mting Physical Intrprtations of Rlativity Thory Moscow 30 Jun-3 July 005 pp [1] I. A. Urusovskii Gravitational Wavs and Papaptrou Copyright 01 SciRs.

8 1756 I. A. URUSOVSKII Mtric in th Si-Dimnsional Tratmnt of Gravitation Physics of Wav Phnomna Vol. 18 No pp doi: /s x [13] A. Papaptrou Ein Thori ds Gravitationsflds mit inr Fldfunrtion Zitschrift fur Physik Vol pp doi: /bf Copyright 01 SciRs.