New perspectives on the classical theory of motion, interaction and geometry of space-time

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1 Nw prspvs on h lassal hory of moon, nraon an gomry of spa-m. R. Hajsfanar Dparmn of Mhanal an rospa Engnrng Sa Unvrsy of Nw York a Buffalo Buffalo, NY 46 US ah@buffalo.u bsra By xamnng h hory of rlavy, as orgnally propos by Lornz, Ponar an Ensn, a funamnal hory of gnral moon s vlop. From hs, h rlaonshp bwn spa-m an mar s sovr. s a rsul, h gomral hory of nraon s nrou. h orrsponng gomral hory of lroynams rsolvs h orgn of lromagn nraon, as a vorx-lk fl, an larfs som of h xsng ambgus.. Inrouon Ponar s hory of rlavy xplans h physal manng of h Lornz ransformaon among nral sysms by unfaon of spa-m. lhough shows a rlaonshp bwn pur Lornz ransformaon an hyprbol roaon, os no spfy wha s roang. hs s h orgn of mos roubls n h hory of rlavy an lroynams. For xampl, alhough h Maxwllan hory of lroynams s h mos unrsoo among h hors of funamnal fors, h lromagn nraon, all h Lornz for, s no a r onsqun of Maxwll s quaons. I has o b posula n an npnn mannr, whh s h manfs of nomplnss

2 of h hory. lhough has bn no ha h lromagn fl srngh nsor an Lornz for ar boh a naural onsqun of h gomr sruur of Mnkowskan spa-m, s funamnal manng has no bn sovr. nohr roubl s h magn monopol whos xsn s apparnly ompabl wh fully symmrz Maxwll s quaons. I sms only mofaon of Maxwll s quaons suff o allow magn hargs n lroynams. Howvr, no magn monopol has bn foun o hs a. o rsolv hs an ohr ffuls, w vlop a funamnal gomral hory of moon an nraon, whh shows ha h Lornz for an Maxwll s quaons ar smpl gomral rlaons bas on four-mnsonal roaon. I s sn ha hs gomry s non-eulan wh nrsng onsquns. hs hory larfs h rlavy of spa-m an s rlaonshp wh mar. I also rvvs h a of h lromagn fl as vorx moon n a unvrsal ny. In h followng son, w frs prsn h hory of rlav nral sysms an knmas of parls n h framwork of Ponar s rlavy. Subsqunly, n Son 3, w vlop h onssn hory of movng parls by xplorng h rlaon bwn mass an spa-m. hs rsolvs h roubls n Ponar s rlavy by larfyng h orgn of h govrnng non-eulan gomry. frwars, n Son 4, w vlop h gomral hory of funamnal nraon, whh shows ha a Lornz-lk for as a roaonal ff s an ssnal harar of vry funamnal nraon. hrfor, vry funamnal nraon s spf by a four-mnsonal vorx-lk fl. Inrsngly, hs mans a unfaon of all fors bas on h gomral hory of moon an nraon. In son 5, w monsra all h als of hs vorx hory for lromagn nraon. hrfor, lroynams s ompl wh lr hargs an magn monopols o no xs. h gomral vw also larfs h spn ynams of harg lmnary parls. h n, s sn ha h orrsponng onssn hory of gravy s a gnralz

3 Nwonan gravy. hs analogous Maxwllan hory of gravy s also vlop n al n Son 6. summary an gnral onluson s prsn n Son 7.. Ponar s hory of rlav nral sysms s an nral rfrn fram n h Mnkowskan spa-m, a four-mnsonal oorna sysm x x x3x4 s onsr suh ha x xx3 s h usual spa an x 4 h axs masurng m wh magnary valus, suh ha four-vor bass (,,, ) (,,, ) x 4. By onsrng h un (.) 3 4 (,,,) (,,, ) h spa-m poson four-vor an b rprsn by Howvr, for smply w somms wr or vn an also ofn us x n pla of x. x x (.) (, ) ( x, y, z, ) x ( x, x 4) x (.3) x ( x, x4) (.4) Wh hs onvnn lmnary noaon, w o no n o us ovaran an onravaran forms of four-nsors n mr noaons. Imporanly, s sn ha h non-eulan gomry govrnng moon an nraon s muh larr n hs omplx numbr noaon. Howvr, all vlop hory an b asly prsn n any ohr noaon. h squar lngh of poson four-vor s x x x x x x x x + y + z (.5) 3

4 whr w no ha h sam symbol x also rprsns h marx form of x. homognous Lornz ransformaon x Λ x (.6) s any ransformaon whh lavs h lngh of h four-vors nvaran x x x x (.7) hs rqurs x x Λ Λ x x (.8) α β α β whh las o h followng orhogonaly onon on Λ Λ Λ δ (.9) α β αβ s wll b sn, w us only frs an son orr hr an four-mnsonal nsors. hrfor, for onvnn w us h marx rprsnaon for hs nsors wh h sam symbol. Bas on hs onvnon, (.9) an b wrn n mor ompa form Λ Λ (.) hs shows ha h Lornz ransformaon s an orhogonal ransformaon n h spf four mnsonal spa-m. Convrsly, any ransformaon, whh sasfs hs orhogonal onon, s a Lornz ransformaon. ll of hs ransformaons form a group n h mahmaal sns. Wha w hav s h rlaon bwn oornas of a pon or vn n wo ffrn four-mnsonal oorna sysms x x x3x4 an x x x 3x 4. On xps ha unrsanng h manng of hs rlaon s rual n vlopng a hory of spa-m an moon. 4

5 .. Spa roaon famlar xampl of a Lornz ransformaon s h rlav spa ornaon of wo oorna sysms wh ommon orgn, whh s spaal roaon. In gnral, for hs ransformaon, w hav Q Λ (.) whr Q s a onsan propr ral orhogonal marx spfyng h spa roaon of h nw rfrn sysm rlav o h orgnal oorna sysm. In hs as, h ransformaon omposs o x Qx (.) s an xampl, for roaon abou h z-axs wh angl φ, w hav osφ snφ Q snφ osφ (.3) In gnral, for roaon abou an arbrary axs no by un vor n wh angl φ, whr φ φn (.4) w hav ) Q δ ε n φ + ( osφ)( n n δ ) j j mj m sn (.5) j j I s onvnn o assoa an an-symmr marx o som axal vor w ( w, w w ) R w fn by w3 w R w w3 w (.6) w w, 3. If G s an arbrary vor, hn w G R G (.7) whh s a rlaon frqunly us n hs arl. hrfor, (.5) an b wrn as w 5

6 ( )( nn ) Q sn φr n + osφ (.8) In rms of lmns Q os nn nn3 φ + ( osφ) n ( ) ( ) nn osφ + n3 snφ nn3 osφ n snφ ( ) ( ) ( ) ( ) ( ) ( ) osφ n3 snφ osφ + osφ n nn3 osφ + n snφ osφ + n snφ n n osφ n snφ osφ + osφ n 3 3 (.9) By usng Cayly-Hamlon horm, an b shown ha Q xp ( ) R φ xp φ3 φ φ φ 3 φ φ xp n3φ nφ n φ 3 n φ nφ n φ (.) Bas on h Eulr horm for h hr-mnsonal moon of a rg boy, vry propr orhogonal marx Q s quvaln o a roaon abou an axs []. hs mans ha h form gvn hr for Q s a gnral form. In pra, h Eulr angls ar wly us o rprsn h roaon marx Q []. I shoul b no ha h rlaons for bas un spa hr-vors ar Q (.) j j j Q (.) wh Q Q QQ (.3) I s obvous ha for h four-mnsonal bas vors w hav j Λ (.4) Λ (.5) I shoul b no ha (,,,) (.6) 4 4 6

7 whh mans h nw m oorna s h sam as h ol on. L h ral orhogonal marx ransformng from x o x b sgna by Q Q x (.7) x an h son orhogonal marx from x o x b Q x Q x (.8) Hn h marx of ompl ransformaon Q from x o x s x Qx (.9) Q Q Q (.3) In gnral h roaons ar no ommuav. In ohr wors, h roaon vors φ φn an φ φn o no follow h Eulan vor summaons. I an b shown ha φ φ φ φ φ os os os sn sn n n (.3) φ φ φ φ φ φ φ sn n sn os n + sn os n + sn sn n n (.3) hs rlaons ar mor onvnnly rv, f a quarnon rprsnaon of roaons or unmoular rprsnaon wh Cayly-Kln paramrs s us []. I s sn ha h summaon of half vor of roaons φ an φ oby h ruls of sphral gomry. h rangl rprsnng hs vors an b onsr as a sphral rangl on a un sphr, wh h angl oppos o llp vor φ gvn by h angl bwn h wo axs of roaon. hrfor, h voral rprsnaon of spaal roaon s govrn by an llp yp of non-eulan gomry. Howvr, for nfnsmal roaons, hs gomry rus o Eulan gomry, whr h nfnsmal roaon vors ommu. 7

8 8.. Boos h ohr mporan form of Lornz ransformaon s a pur Lornz ransformaon or boos spf wh rlav vloy v. h boos paramr or rapy s fn by v anh (.33) h nvrson of hs rlaon gvs ln anh v v v v v v v (.34) h vor rapy also an b onsr as (.35) whr s h un vor n h ron of v. Hr, w mphasz h us of rapy as an ssnal paramr. smpl xampl of a boos s h boos along h x-axs, for whh osh snh snh osh Λ (.36) whh s usually wrn as γ βγ βγ γ Λ (.37) wh v / β an / ) ( β γ, whr γ osh (.38) βγ snh (.39)

9 h sruur of hs ransformaon nsor s rmnsn of a roaon nsor, bu wh hyprbol funons nsa of rular. Inrsngly, w an fn ψ (.4) whr snψ snh (.4) hrfor, h ransformaon marx an b wrn as os ψ osh (.4) osψ snψ Λ (.43) snψ osψ On an ralz hs s a roaon wh magnary angl ψ rprsnng h vaon of plan x x rlav o plan x x 4 4. nalogous o h spaal roaon, h bas fourvors of h nw sysm ar ( osh,,, snh ) (,,, ) (,,, ) 3 (.44) (,,, osh ) 4 snh hrfor, w hav (, ) osh osψ os (, ) snh snψ os 4 (, ) snh snψ os 4 (.45) (, ) osh osψ os 4 4 Rlaons (.45) show ha hs magnary an omplx angls ar ( ) ψ, 9

10 π π, 4 + ψ + ( ) π π 4, + ψ + (.46) ( ) ( ), 4 4 For a gnral boos, whh s no paralll o any of oorna axs, w hav + (osh ) snh Λ (.47) snh osh By usng h Cayly-Hamlon horm, w an show Λ xp + snh xp + ( osh ) (.48) In rms of h lmns w hav + ( osh ) ( osh ) ( osh ) ( osh ) + ( osh ) ( osh ) ( osh ) ( osh ) + ( osh ) 3 snh 3 snh Λ (.49) snh snh snh 3 snh osh hrfor, w xp h bas four-vors of h nw sysm n rms of ol ons o b ( + (osh ), (osh ), (osh ) 3, ) ((osh ), + (osh ), (osh ) 3, ) ((osh ), + (osh ), + (osh ), ) snh snh (.5) snh 4 ( snh, snh, snh 3,osh ) ( snh, osh ) 3

11 I s sn ha h angls among nw an ol axs an b oban asly. For xampl, from w oban (, ) osh os 4 4 (.5) ( ) ψ (.5) 4, 4 hs shows h angl bwn h m axs s spf by rapy, whh s xp..3. Gnral Lornz ransformaons Evry homognous Lornz ransformaon n gnral an b ompos no a pur Lornz ransformaon Λ B (boos) an a spaal roaon Λ R (n hr orr) []. For h as whr a Lornz ransformaon s rprsn as h prou of a boos + (osh ) snh Λ B (.53) snh osh from h ol sysm x x x3x4 o h nrma sysm y y y3 y4, whr y Λ x (.54) follow by a spaal roaon Q Λ R (.55) from y y y3 y4 o h nw sysm x x x 3x 4 B R w hav h oal homognous Lornz ransformaon whh s x Λ y (.56) Λ Λ Λ (.57) R B Q + (osh ) Λ snh Q + (osh ) Q snh snh osh snhq osh (.58)

12 I s obvous h ransformaons Λ R an Λ B ar no gnrally ommuav. hs s baus h vors an φ ar non-eulan an hrfor hr aon os no follow h ruls of Eulan gomry. Now w monsra h mporan propry of a pur Lornz ransformaon or boos whh follows h hyprbol yp of non-eulan gomry. L h pur Lornz ransformaon from x o x b sgna by Λ x (.59) x Λ an a son Lornz ransformaon from x o x b Λ x Λ x (.6) Hn h marx of ompl ransformaon Λ from x o x x Λ x (.6) s Λ Λ Λ (.6) whr Λ + (osh ) snh snh osh + (osh ) snh snh osh (.63) I s sn ha h ompl ransformaon s no n gnral a pur Lornz ransformaon. hs ransformaon s n gnral form (.58), whr osh osh + (.64) osh snh snh hs rsul s h naon of hyprbol gomry govrnng h vloy aon law. hs has bn no an vlop xnsvly by arly nvsgaors of rlavy suh as Varǎk [-4]. I s sn ha hs non-eulan gomry s h orgn of h famous homas-wgnr roaon, whh has bn xplan by Borl [5]. n aoun of hs

13 nvsgaons an b foun n h arl by Walr [6]. On an ralz ha for nfnsmal rapy vors, h hyprbol gomry rus o Eulan gomry, whr h rapy or vloy vors ommu. If h ransformaons Λ an Λ ar gnral Lornz ransformaons, s sn ha Λ Q whr + (osh ) Q snh snhq osh Q osh snh snh + (osh ) Q snh snh Q osh (.65) osh osh + Q (.66) I wll b shown ha hs rlaon an b furhr gnralz o alrang sysms. W an s ha h nral sysms ar orn from ah ohr by a four-mnsonal roaon. h homognous Lornz ransformaon jus spfs hs roaon rlav o a fx nral sysm as rfrn fram. hs ransformaon n gnral an b ompos no a pur Lornz ransformaon (boos) an a spaal roaon. In a gomral vw, h Lornz ransformaon an b spf by a hyprbol vor rprsnng h hyprbol angl assoa wh h boos an an llp vor φ rprsnng h spa angl roaon. h gomry govrnng hs vors s non- Eulan as was monsra. In gnral, h bas un four-vors of wo nral sysms ar rla by Λ (.67) or Λ (.68) hrfor, h angls among hs rons ar suh ha (, ) Λ os (.69) 3

14 (, ) Λ os (.7) I woul b nrsng o prsn a smpl gnral Lornz ransformaon. L hs ransformaon b h prou of a boos n h x-ron follow by a spaal roaon aroun a z-axs osφ snφ osh snh snφ osφ Λ (.7) snh osh whh an b wrn as osφ osh snφ osφ snh snφ osh osφ snφ snh Λ (.7) snh osh h bas un four-vors of h nw sysm ar ( osφ osh, snφ,, osφ snh ) ( snφ osh, osφ,, - snφ snh ) (.73) 3 (,,, ) (,,, osh ) 4 snh I s no ha (, ) osφ osh os (, ) snφ os (.74) (, ) os 3 (, ) osφ snh os 4 I s sn ha hs rlaons ar h rsul of h aon of non-ommuav non- Eulan vors an φ. 4

15 Wha w hav monsra s h vry mporan harar of h s of four-mnsonal sysms wh hr ral oornas an on magnary oorna. I s sn ha hs sysms ar orn from ah ohr, n a mannr whh an b rprsn by a ombnaon of rular an hyprbol angls. I an b ralz ha hs s s h s of all nral sysms n Ponar s rlavy. In hs hory, spa an m ar no longr spara as n Gallan rlavy an moon s nohng bu roaon. Howvr, Ponar s rlavy os no spfy wha s roang. Our am n h followng sons s o rsolv hs funamnal quson. lhough w hav bn usng h onp of four-vor an four-nsor rpaly, w hav no gvn hr rgorous fnon. hrfor, for fuur rfrn, h fnon of a four-nsor s prov hr. four-nsor G of orr n s fn as a mahmaal obj wh n ns whh has 4 n omponns ransforms va G L n L Λ G n Λ LΛ n n n a gvn nral sysm an G (.75) o a nw nral sysm. h mos mporan four-nsors ar hos nvolv n h hory of lroynams, whh wll b suss lar. For smply, w hav bn usng h sam symbols suh as x,, Q, x, an Λ o rprsn h marx form of hr orrsponng hr an four nsors. L n 3. Funamnal hory of moon In hs son, w vlop h hory of alrang parls, whh shows h funamnal rlaon bwn spa-m an mar. I larfs h rlavy of spam an shows how an nral sysm ransforms o ohr nral sysms. hs s nohng bu h gomral hory of nraon. I s sn ha h rlav moon s h rsul of h four-mnsonal roaon of hs sysms rlav o ah ohr. W sar wh lassal parl knmas an vlop h funamnal hory of moon. h hory of nraon wll b suss n h nx son. 5

16 3.. Knmas of a parl L us spfy h nral sysm x x x3x4 as an nral rfrn fram. Consr a parl wh mass m movng rlav o hs nral fram. any poson, h moon may b onsr as akng pla n h plan ha onans h pah a hs poson. hs plan s ofn all h osulang plan. h vloy vor v s angn o h pah urv n hs plan. h alraon of h parl v a (3.) ls also n hs plan. W an onsr a loal oorna sysm by fnng h un vor angn o h urv a hs poson, h un vor n n h ron of prnpal normal o h urv n h osulang plan, an h b-normal un vor b, whh s normal o h osulang plan a h pon. h rlaon (3.) b among hs vors hols. In hs loal (angnal, normal, b-normal) oorna sysm w hav v v (3.3) wh ss v (3.4) whr s s s h lngh of h nfnsmal splamn of h parl on h spa urv n m nrval. For alraon, w hav v a + v (3.5) For h son rm, w apply h onp of urvaur n h form s s n n (3.6) s R whr R s s h raus of urvaur a h parl poson pon. hrfor, h alraon n rms of angnal an normal omponns a an a n s a a + a n (3.7) 6

17 whr v a a (3.8) a n v ann n (3.9) R s I sms h ffrnal gomry govrnng h knmas of h parl s mor ompl f w nrou h onp of orson of h urv fn by b n (3.) s R whr R or s h raus of orson of h urv. I an b asly shown ha s s s or n n + b (3.) s R R or hrfor, h quaons for urvaur of urv mgh b wrn as ss n s S Rs b ss R s R or R or n b (3.) hs rlaon s all h Frn-Srr formula n ffrnal gomry. h ansymmr nsor on h rgh han s posssss h whol nformaon abou h urvaur an ws of h urv a h pon unr onsraon. Howvr, an nrsng nrpraon of hs rlaon an b gvn as follows. h prnpal rons spfy a loal orhogonal rfrn sysm aah o h parl. hs rfrn sysm roas as h parl movs on h urv pah. I s obvous hs rlaon shows h graual roaon of hs loal sysm wh rsp o any nral sysm. If w wr h rlaon as n b 7

18 8 b n or or s s b n R v R v R v R v (3.3) h an-symmr nsor s h angular vloy nsor of h roang loal sysm b n. By onsrng h angular vloy vor s or R v R v ω (3.4) w oban h rlaon b n b n ω (3.5) I s nrsng o no ha hr s no angular vloy omponn n h n ron. h Frn Srr formulas an b gnralz o hghr mnsonal Eulan spas by fnng gnralz urvaurs. I an b shown ha n h prnpal loal oorna sysm, whh s all h Frn Srr fram, h an-symmr urvaur nsor s r-agonal [7]. n mporan analogy wll b sn n vlopng h rlavs hory of moon.

19 3.. Rlavs knmas of a parl In a rlavs suy, h vloy an alraon of h parl mus b fn as four-vors. Howvr, s sn ha h vors v an a ar sll usful n hs vlopmn. h poson of a parl n h nral rfrn fram srbs a pah known as h worl ln. By onsrng wo nghborng vns on h worl ln of h parl wh oornas x an x + x, w hav ( x, ) ( v ) h squar lngh of hs nfnsmal four-vor x, (3.6) v s x x x x x (3.7) s h salar nvaran unr all Lornz ransformaon. I s sn ha h magnary lngh on h worl ln s v s (3.8) h propr m bwn h vns τ s fn by hrfor, v τ γ (3.9) s τ (3.) By usng h onp of rapy w no v anh (.33) γτ τ osh (3.) 9

20 h un four-vor angn o h worl ln s fn as whr x x (3.) s s (3.3) h four-vor vloy u u s fn as h ra of hang of h poson vor of parl x wh rsp o s propr m x u (3.4) τ h spa an m omponns of u ( ) u u,u 4 (3.5) ar an u γv v v snh (3.6) u γ 4 v osh (3.7) hrfor whh an b wrn as wh ( snh, osh ) u (3.8) u (3.9) ( snh, osh ) (3.3) h lngh of h four-vor vloy s a onsan sn an s hus m-lk. I s sn u u uu u + u4 (3.3)

21 (3.3) s whh mans s s normal o h worl ln. By onsrng h un four-vor n n hs normal ron all h frs normal an usng h onp of urvaur, w hav n (3.33) s R whr R s h worl ln raus of urvaur a h pon unr onsraon. h mnus sgn s for onvnn an wll b jusf shorly. I s sn ha (3.34) s s R h four-alraon b b s fn as u x b (3.35) τ τ whh s always prpnular o h four-vor vloy whr, u u (3.36) τ I an b asly shown ha 4 v a 4 v a b γ a + γ v, γ (3.37) h lngh of four-vor alraon an b foun o b 4 6 v a b bb γ a + γ (3.38) Sn b b s posv, h four-alraon s spa-lk. Howvr, s mor appalng o onsr h four-alraon rlav o h worl ln. By usng (3.33), w oban b τ R n (3.39)

22 whh shows h four alraon s n ron of frs-normal of worl ln an s valu s b (3.4) R Now w suy h rlav moon of a parl n ffrn nral sysms. L us onsr h nral sysms x xx3x4 an x x x 3x 4, whh ar rla by h Lornz ransformaon x Λ x (.6) whr Q + (osh ) Λ snh Q + (osh ) Q snh snh osh snhq osh (.58) ssum parl movs n h frs nral sysm wh ( ) vloy s whr u x an s four-vor x ( u, u 4 ) ( snh, osh ) (3.4) v anh (3.4) hs parl also movs n h son nral sysm wh x x ( ) (, u ) ( 4 snh, ), suh ha u u osh (3.43) whr v anh (3.44) hs four-vor vlos ar also rla by h nsor ransformaon hrfor, u Λ u (3.45)

23 Q + (osh ) Q snhq snh u (3.46) snh osh osh whh gvs h rlaons snh Q[ + (osh ) ]snh snh osh Q (3.47) osh osh osh snh snh (3.48) h rlaon (3.48) shows ha h vloy aon law s val vn whn on of h vlos s no onsan. W nvsga shorly h valy of hs law whn all parls ar alrang Moon of parl as a four-mnsonal roaon fr rvwng knmas of a parl, w vlop h mporan harar of s moon as a four-mnsonal roaon. o show hs w onsr h moon of h parl as h ransformaon of s four-vloy vor u n h nral rfrn fram sysm. L u b h nal four-vor vloy a poson x, suh ha u ( x ) u. W an onsr h ransformaon u ( x) L ( x, x) u ( x) (3.49) whr h ransformaon nsor L ( x, x ) pns on h urrn poson of h parl. hs rlaon an b wrn as ( ) L( x, x u u x ) (3.5) Sn h lngh of h four-vor vloy s onsan, w hav u ( x) u ( x) u ( x ) u ( x) (3.5) hrfor, Lα ( x, x ) Lβ ( x, x) δ αβ ] uα ( x) uβ ( x ) (3.5) [ hs rqurs h orhogonaly onon L ) δ (3.53) α ( x, x) Lβ ( x, x αβ 3

24 I s sn ha alhough L( x, x ) looks smlar o a Lornz ransformaon among nral sysms, vars wh moon of h parl. Manwhl, h nvrs rlaon s whh n rms of omponns s ( x ) L x, x u( x) u ( ) (3.54) u ( ) L x, x u ( x) x (3.55) ( ) hs gvs h orhogonaly onon n h form L x, x ) L ( x, ) ( x (3.56) or L ) δ (3.57) α ( x, x) Lα ( x, x By akng h rvav of (3.57) wh rsp o h propr m of h parl, w oban L α ( x, x) Lα ( x, x) Lα ( x, x) + Lα ( x, x) τ τ (3.58) Now by fnng h four-nsor Ω Lα ( x, x) x) Lα ( x, x ) (3.59) τ ( w an s ha h rlaon (3.58) boms Ω ( x) + Ω ( x) (3.6) or Ω ( x) + Ω ( x) whh shows Ω ( x) s an an-symmr four-nsor. In ompa form, w hav (3.6) L( x, x) Ω ( x) L ( x, x) (3.6) τ 4

25 By mulplng h rlaon (3.59) wh L β x, x ) an usng h orhogonaly onon, ( w oban L ( x, x τ ) Ω (3.63) α ( x) L ( x, x ) α hs may raly b wrn as L( x, x τ ) Ω ( x) L( x, ) x (3.64) Now h alraon from h orgnal ransformaon rlaon s u ( x) L ( x, x u ( x ) u ) (3.49) ( x) τ By subsung from (3.63), w hav u τ whh rus o h rlaon ( x) u L (, ) x x u ( x ) (3.65) τ ( x) L ( x, x u ( x ) Ωα α ) (3.66) τ ( x) ( x) u ( x) Ω (3.67) α α hs s h rlaon bwn four alraon u τ ( x) an four-vloy a ah pon on h worl ln. I shoul b no ha h rlaon (3.67) s aually (3.35) an (3.37) wrn as a ransformaon. I s also no ha h rlaon (3.67) s smlar o h non-rlavs rlaon for ra of hang of a onsan lngh vor G aah o a roang sysm G ω G (3.68) 5

26 whr ( ω, ω ω ) ω s h angular vloy of ha roang sysm. I s rmmbr, 3 ha h omponns ω, ω an ω 3 ar h angular vlos of h boy sysm n h yz, zx an xy plans of an nral fram. Baus of h mporan of (3.68), s avanagous o monsra h mahmaal als of s rvaon. L h prm sysm o b h boy sysm. hn h omponns of hs vor G ar onsan n hs sysm. hrfor, w hav G Q( ) G( ) (3.69) whr Q () s h orhogonal roaon marx. hs rlaon an b wrn as ( ) Q ( )G G (3.7) h ra of hang of h vor G ( ) rlav o h fx rfrn fram s G Q fr lmnangg by usng (3.69), w oban ( ) G (3.7) Now by fnng h nsor G Q QG (3.7) w hav G WG whh n h nx noaon an b wrn as Q W Q (3.73) (3.74) G WjG (3.75) j Now by ffrnang h orhogonaly onon wh rsp o h m, w oban Q Q (3.76) 6

27 Q Q + Q Q (3.77) whh may raly b wrn as W + W (3.78) hs rlaon shows ha h nsor W s an-symmr. hs nsor s h known angular vloy nsor of h roang sysm rlav o h nral sysm. In rms of lmns, hs nsor s ω3 ω W R ω ω3 ω (3.79) ω ω hrfor, h rlaon (3.74) s h ohr form of (3.68) G ω G (3.68) I shoul b no ha h Frn-Srr formula (3.5) s h applaon of hs quaon for funamnal bas un vors. Now, w hav a rmarkabl analogy for u τ wh ( x) ( x) u ( x) Ω (3.67) α α G WjG (3.75) j u I s sn ha h four-vor alraon s h rsul of onnuous roaon of h τ four-vor vloy u n a four-mnsonal sns. hrfor, sms u s aah o a four-mnsonal sysm x x x 3x 4 n h x 4 ron, whr (,, ) u, (3.8) 7

28 an hs sysm s roang wh four-mnsonal angular vloy Ω rlav o h nral sysm, suh ha L ( x, x τ ) Ω (3.63) α ( x) L ( x, x ) α hrfor, w hav sovr ha hr s a funamnal rlaon bwn spa-m an mar. massv parl spfs a loal four-mnsonal orhogonal sysm wh hr ral axs an on magnary axs. Whn hs loal x x x 3x 4 spa-m sysm, h parl has aah four-vloy wh magnu n h m ron. h roaon of hs spa-m or four-mnsonal sysm gnras moon of h parl rlav o h nral sysm. hs roaon s rprsn by h four-nsor angular vloy Ω Ω n h nral rfrn fram. h naur of hs four-mnsonal angular vloy s xplor vry shorly. s was mnon abov, a any pon on h worl ln, w hav h ransformaon ( x) u ( x) u Λ (3.8) whr h varyng nsor ransformaon Λ ( x) looks lk a Lornz ransformaon. hrfor, hr mus b a rlaon bwn nsors Λ ( x) an ( x). For a parl a L an nal pon x, w hav ( x ) u ( ) u Λ (3.8) x hrfor Λ ( x) u ( x) Λ ( x ) u ( ) x (3.83) By subsung for u ( x) from (3.49), w oban Λ ( x) L ( x, x u ( x ) Λ ( x ) u ( x ) α ) α α α (3.84) hs shows h rlaon 8

29 whh an b wrn as ( x) L ( x, x Λ ( x ) Λ α ) α (3.85) ( x) ( x, x Λ( x ) Λ L ) (3.86) hrfor ( x) Λ( ) L ( x, x) Λ x (3.87) I shoul b no ha alhough Λ ( x) s no onsan any mor, follows h gnral form of Lornz ransformaon (.58) QP + (osh ) PP snhpp Λ snhpp oshp (3.88) Q + P (oshp ) QPPP snhpqpp snhpp oshp whr h physal manng of h paramrs n P an Q P has no bn spf. Howvr, w an xplor hr rlaon wh h moon of h parl n h ours of our vlopmn. By usng h rlaon (3.8), w oban u ( snh, osh ) (3.89) P P P hs shows h vor P s aually h rapy vor of h parl. hrfor, Q P + (osh ) Λ snh Q P + (osh ) Q P snh snh osh snhq P osh (3.9) I s sn ha h poson vor x( ) x of h parl os no spfy s rlav poson n h rfrn nral fram omplly. I s also nssary o spfy s boy fram ornaon Λ() Λ rlav o hs fram. Howvr, h rapy vor () s oban from h vloy vor () an orhogonal marx ( ) x v. hrfor, h poson vor x x( ) Q omplly spfy h parl poson. P Q P 9

30 Now, w nvsga h harar of h an-symmr nsor Ω Ω. h nal four-vor vloy s ( snh, ) osh u (3.9) For smply w ak h spa oornas of h nal boy fram o b paralll o h saonary nral fram, whr Q ( ) P. hrfor, + (osh ) snh Λ (3.9) snh osh By akng h rvav wh rsp o h propr m τ n h quaon w oban ( x) Λ( ) L ( x, x) Λ x (3.87) L( x, x) Λ τ τ ( x) Λ ( x ) (3.93) hrfor, h rlaon boms an fnally w hav ( x) L( x, x) Ω ( x) L ( x, x) (3.94) τ ( x) Λ Λ( x ) Λ ( x ) Λ( x) (3.95) τ Ω Ω ( x) ( x) Λ Λ( x) (3.96) τ For ( x) Λ from (3.9) 3

31 3 + osh snh snh ) (osh P Q Λ (3.97) By akng h rvav wh rsp o h propr m, w oban osh snh snh ) (osh snh snh osh snh osh ) (osh snh τ τ τ τ τ τ τ τ P p Q Q Λ (3.98) hrfor, for ( ) x Ω n (3.96), w hav τ τ τ τ τ τ τ osh snh snh ) (osh osh snh snh ) (osh osh snh snh ) (osh snh snh osh snh osh ) (osh snh P P Q Q Ω (3.99) Now by usng h rlaon P P P Q Q R ω τ (3.) whr P ω s an angular vloy vor n a mahmaal sns, w oban

32 snh snh snh snh ) (osh P P P ω ω ω R R R Ω τ τ (3.) hs rlaon an b wrn n h form η η R Ω ω (3.) whr + P R R ω ω ) (osh (3.3) + P R η ω τ τ snh (3.4) On an s hs rlaons an also b wrn as ( ) P ω ω + τ osh (3.5) + P ω η τ τ snh (3.6) hs rlaons an b smplf furhr by usng h rlaons n s R v τ osh (3.7) an n b (3.) h four-nsor ( ) x Ω n rms of lmns n h nral rfrn fram s η η η η ω ω η ω ω η ω ω Ω (3.8)

33 hs s h gnral form of an an-symmr four-nsor angular vloy Ω. I shoul b no ha h lmns Ω4 Ω4 η ar magnary. I s obsrv ha h angular vlos ω, ω an ω 3 n xy, yz an zx plans gnra spa roaon of h boy fram; h magnary angular vlos η, η an η3 n x, y an z plans gnra boos of h boy fram. hrfor, h spa-m boy fram sysm roas rlav o h nral sysm wh angular vloy nsor Ω, whh s a ombnaon of llp an hyprbol angular vlos ω an η. Rurnng o h quaon for four-alraon ( x) u Ωα ( x) uα ( x) (3.67) τ w hav h spa an m omponns of four-vor alraon as u ω u τ η u 4 η u τ u 4 (3.9) (3.) hs rlaons an also b wrn n h form v v / u η + ω v (3.) u 4 η u v / (3.) o monsra h physal manng of h four-nsor angular vloy Ω, w onsr h as whr h parl sars movng from rs a. hs rqurs n (3.6) an (3.7). hrfor, a hs momn, ω ω P (3.3) 33

34 η τ a (3.4) hs man ω an η a ar h rular an hyprbol angular vloy of h boy fram rlav o h nral fram. W no ha a hs nsan τ an η a a (3.5) an hrfor φ ωτ (3.6) v aτ aτ (3.7) h nfnsmal an-symmr four-mnsonal roaon nsor Φ s fn whh an b wrn as R φ Φ v Φ Ωτ (3.8) v (3.9) hs nsor n rms of lmns s φ3 φ v φ3 φ v Φ (3.) φ φ v3 v v v3 hs xplanaon an b us for h spal as whr h nral sysm s onn wh h boy fram nsanly, whh s ofn all a ommovng nral fram sysm. For hs as, w hav Q P an h rlaon boms L τ L ( x) τ Ω ( x) α Ω ( x) L ( x) α ( x) (3.) (3.) 34

35 whr Rω Ω η η Rω a a (3.3) I s sn ha an a hs nsan ω (3.4) ω P η a a (3.5) τ an w hav φ ω τ (3.6) v a τ a τ (3.7) I s sn ha ω an η a ar h rular an hyprbol angular vloy of h boy fram rlav o h ommovng nral fram sysm. h nfnsmal fourmnsonal roaon Φ of h boy fram rlav o h ommovng nral fram sysm s R φ v Φ (3.8) v I s obvous h four-nsor angular vloy nsor h ommovng fram an w hav Ω s h rprsnaon of Ω on Ω Ω (3.9) From hs, s xp ha Ω Λ α Λ (3.3) βω αβ hs nsor ransformaon an also b wrn as Ω L L Ω α β αβ (3.3) 35

36 lhough w sll us h noaons ω an η an all hm angular vlos, hs vors anno b akn as a propr angular vloy vors lk vors ω an η. hs s h rsul of h non-eulan gomry govrnng h four-mnsonal roaons. ombnaon of h rular an hyprbol angular vlos ω an η n h rlaon (3.3) gvs h vors ω an η. h famous homas prsson for alrang parls s manfs of h govrnng hyprbol gomry. Now s lar why w no subsrp P n h orhogonal nsor orhogonal nsor Q spfs ω hrough h rlaon Q P, whh spfs ω P. h R ω Q Q (3.3) lhough Q an ω ar ssnal mahmaal ns, hy anno b monsra gomrally as rly as Q P or ω P. Howvr, w mus b arful whn w onsr ω as a rular angular vloy. W mgh rop h subsrp P auously. hrfor, w hav larn ha h moon of a parl n h lassal sns s h rsul of h hyprbol par of roaon of s boy fram. h spa roaon s also par of h moon, whh s h orgn of spn prsson of an lron n a magn fl. hs wll b suss n mor al shorly. I s ralz ha h non-eulan gomry s h rsul of ransformng four-nsors an four-vors among ffrn spa-m boy frams. hrough hs mporan physal raly, on appras h work of hos who onsr h possbly of non- Eulan gomry. h non-eulan asp of h vloy aon law for unform moon has bn su by Robb, Varǎk, Lws, Wlson an Borl [6]. Howvr, hs sovrs hav no bn appra nough by lar nvsgaors. Forunaly, hr hav bn som avoas of rvvng hs mporan ssu rnly [8]. Now w appra ha hs pah rsolvs nonssns an paraoxs n rlavy. I also 36

37 xplans h gomral mhansm bhn moon an nraon, whh wll b vlop n h nx son. W hav also no an mporan ssu rgarng h four-vor vloy of a parl. I has bn shown ha h four-vor vloy s aah o s boy fram suh ha ( x) u ( x) u Λ (3.8) hs has bn shown symbolally n Fg. by onsrng a wo mnsonal spa an on m ron. I shoul b no ha h nral rfrn fram an boy fram of h parl boh hav aah four vor-vlos u R an P u n hr spa-m frams, rspvly. Howvr, h Lornz ransformaon (3.8) rlas h omponns of four-vor vloy u of parl P n s fram an s omponns of four-vor vloy u ( x) n h nral rfrn fram of parl R. I shoul b no ha h four-vor vloy omponns u (,,,) an u ( x) ( snh, osh ) ar rprsnaons of P u n boy fram of parl an nral rfrn fram, rspvly. Inral rfrn fram Boy fram of parl x x x x R u u ( x) ( snh, osh ) x 4 P u (,, ) u, x 4 Fg.. Inral rfrn fram an boy fram. hrfor, w an onsr a nw yp of four-vor G all an aah four-vor an fn as a four-vor aah o h boy fram of a parl, suh ha G Λ G (3.33) no mar whhr h boy fram s nral or alrang. For hs four-vor 37

38 G τ Ω G (3.34) hs an b wrn as or n ompa form G s Ω G (3.35) G ΩG (3.36) s Usng hs rlaon for un bas angnal four-vor, w hav Ω (3.37) s By omparng hs rlaon wh h rlaon (3.33) for h worl ln raus of urvaur, w oban n Ω R I s sn ha h worl ln raus of urvaur sasfs whr h symmr nsor Ω Ω s R (3.38) Ω (3.39) ωω ω + ηη ω η ( ) ω η η (3.4) I s also sn ha h funamnal quaon (3.63) an b wrn as L s or n h ompa form ( x) ( x) L s Ωα ( x) Lα ( x) (3.4) Ω( x) L( x) (3.4) 38

39 39 For h bas four-vors of boy fram, w hav ( ) x s Ω (3.43) whh an b wrn as s s s s η η η η ω ω η ω ω η ω ω (3.44) On ralzs ha hs quaon s aually a Frn-Srr-lk formula for ornaon of h loal boy fram rlav o h nral sysm. Howvr, shoul b no ha hs ornaon s n rms of gnralz urvaurs of h worl ln bu no gnrally n prnpal rons. h angn o h worl ln spf by 4 s a prnpal ron, bu h prpnular rons o h angn ar no usually prnpal rons. I shoul b also mnon ha Syng has alray su h Mnkowskan Frn-Srr movng fram [9]. Wha w hav shown s ha hs fram s a rprsnaon of h funamnal boy fram of a parl. In hs son, has bn monsra ha hr s a rlaonshp bwn Mnkowskan spa-m an massv parls. h parl spfs s spa-m boy fram rlav o h nral rfrn fram. Now h naural quson onrns h vry xsn of hs spa-m sysms. I s sn ha w ar ompll o am h xsn of a unvrsal ny, whh has nohng o o wh any spal spa-m. I s n hs unvrsal ny n whh parls an hr orrsponng spa-m boy fram xs. Lar w wll nvsga mor abou hs unvrsal ny.

40 3.4. Gnral rlav moon an vloy aon law Now w vlop h hory of rlav moon for gnral alrang parls. I s sn ha h govrnng rlaons an vloy aon law n Ponar s rlavy ar sll val for hs gnral as. Consr wo parls an B movng wh vlos () v an v ( ) v B v B rlav o an nral sysm. h four-vor vlos u an u B ar aah fourvors, whr w hav ( u ) Λ u (3.45) ( u ) an ( B ) B whr ( ub ) Λ BuB B (3.46) u ar rprsnng hs four-vors on hr orrsponng boy fram ( ) ( u ) (,, ) u (3.47) B, B h ransformaons () Λ an ( ) Λ Λ rprsn h ornaon of hs boy B Λ B frams rlav o h nral fram. For hs ransformaons, w xplly hav an Q + (osh ) Q snh Q Λ () (3.48) snh osh Q B + (oshb ) Q BB snhbq BB Λ () B B (3.49) snhbb oshb By usng (3.47) an ombnng (3.45) an (3.46), w oban u Λ Λ u (3.5) B B Rlav ornaon of h boy fram B rlav o a m s no by Λ an s B fn suh ha Λ Λ Λ (3.5) B B 4

41 hs rlaon shows Λ Λ Λ (3.5) B B hrfor, (3.5) boms u Λ u (3.53) B B whh an also b wrn as u B Λ u (3.54) B I shoul b no ha Λ s h rlav Lornz ransformaon from boy fram B o boy fram B masur by our nral rfrn fram a m. hrfor all h rlaons ar rlav o hs obsrvr a m. Howvr, w shoul rv smlar rlaons rlav o h obsrvr aah o h boy fram. For hs w no ha h vloy of B rlav o masur by an obsrvr n h boy fram of s ( ub ) ( ub ) Λ ub (3.55) By subsung for u B from (3.46), w oban ( ub ) ( ub ) Λ Λ B ( u B ) B (3.56) W also hav h obvous rlaon whh an b wrn as ( u ) ( Λ B ) ( ub ) (3.57) ( ub ) ( Λ B ) ( u ) (3.58) By omparng (3.56) an (3.57) an usng (3.55) w oban h rlaon whh an b wrn as ( ) Λ Λ Λ (3.59) B ( ) Λ Λ B B B Λ (3.6) 4

42 Inrsngly, s sn ha ( ) Λ Λ Λ Λ (3.6) B B whh looks lk h ransformaon for nsor Λ from nral rfrn fram o h B boy fram. Wha w hav s h vlopmn of h gnral hory of rlav moon. Explly from (3.55), w hav snh u B (3.6) snh osh oshb Q + (osh ) Q snh Q B B ( ) From hs, w oban h rlaons ( snh B / B ) snh osh BQ + Q [ + (osh ) ] snh BB ( B ) osh oshb snh snhb B (3.63) osh (3.64) hs rlaons ar h manfs of hyprbol gomry govrnng h vloy aon law vn for alrang parls. hs propry hols for all aah four-vors an four nsors. Inral obsrvrs rla omponns of aah four-vors an fournsors by Lornz ransformaons. hs s h orgn of non-eulan gomry govrnng h hr vor an hr nsors. s w saw h aon of hr vor vlos follow hyprbol gomry. I shoul b no ha hs rlaons hol sp h fa ha h ransformaon x Λ x (3.65) s no val among alrang sysms. Wha w hav hr s h omplon of h Ponar s rlavy for alrang sysms. 4

43 4. Funamnal nraon fr vlopng h hory of alrang moon, w ar ray o vlop h hory of funamnal nraon. h quaon of moon for a parl n an nral rfrn fram sysm s gvn by u m F (4.) τ whr F s h four-vor Mnkowsk for. hs for s h rsul of nraon of h parl wh a fl, suh as an lromagn fl. W ar lookng o xplor h gomral harar of hs fl. By subsung for four-alraon from (3.67) n h rlaon (4.), w oban for h Mnkowsk for. Sn F mω u (4.) Ω s an-symmr, w hav F u mω u u (4.3) whh mans h four-vor Mnkowsk for F s prpnular o h four-vor vloy u. h rlaon (4.) shows ha hs for pns on four-vor vloy u an four-nsor angular vloy Ω a h poson of h parl ~ x. s a rsul, h fl srngh mus pn on h four-nsor angular vloy Ω. I s sn ha h smpls amssbl fl s hararz by a fl srngh four-nsor ( x) Θ suh ha a h poson of h parl ( x ) mω ~ (4.4) αθ Salar α s a propry of h parl an pns on h yp of nraon. hs quany an b rognz as lr harg n lromagn nraon. hrfor, w an onsr a funamnal nraon o b an nraon hararz by an an- Θ x, suh ha a h poson of h parl ~ x symmr srngh nsor fl ( ) α Ω ( ~ Θ x ) (4.5) m 43

44 lhough ( x~ ) Θ s npnn of h parl, h Mnkowsk for pns on h parl hrough α an four vor vloy u, suh ha F αθu (4.) hrfor, h quaon of moon boms u ( x ) u (4.6) τ m αθ ~ On an s ha h an-symmr srngh nsor ( x) Θ looks lk a four-mnsonal vory fl analogous o h hr-mnsonal vory n roaonal flu flow. hrfor, w an onsr a four-vor vloy-lk fl V V nu o h spa-m of h nral rfrn fram, suh ha s four mnsonal url s h vory-lk srngh nsor ( x) V V Θ (4.7) From our famlary wh lroynams, s obvous ha lromagn nraon s omplly ompabl wh hs gomral hory of nraon. hrfor, n h nx son, w prsn h ovaran hory of lromagns an xplor s gomral asps bas on h four-mnsonal vory hory. I s sn ha hs gomral hory rsolvs som ambgus n h raonal hory of lromagns. Mor mporanly, on ralzs ha hs hory s a mol for any ohr funamnal nraon. hrfor, h orrsponng gravaonal hory s also vlop n al n Son 6. W shoul rmmbr ha h hory of rlavy has s orgn n h hory of lroynams. Now w an s ha h hory of nraon also has s orgn n hs hory. 44

45 5. Gomral hory of lromagn nraon In h hory of lroynams [], n an nral rfrn fram, h for on a harg parl an b xprss n rms of wo vor fls, an lr fl E(x, ) an a magn fl B(x, ). In rms of hs fls, h for on a parl wh harg q movng wh vloy v s gvn by F q( E + v B) (5.) q( E B v) hs s known as h Lornz for n SI uns. I s no ha h vor B s aually an axal or psuo-vor. hrfor, hr s a orrsponng an-symmr nsor B3 B R B B3 B (5.) B B suh ha h Lornz for n marx form s F q ( E R v) (5.3) B In h ovaran hory of lroynams, h orrsponng four-vor Mnkowsk for s F qf ~ x u (5.4) whr h lromagn srngh fl F s ( ) B3 B E B3 B E R B E F (5.5) B B E3 E E E E3 hrfor, h quaon of moon of hs parl s gvn by u ( x ) u (5.6) τ m qf ~ 45

46 I s obvous ha h quaon (5.6) has h form of h quaon (4.6), whh was oban bas on h knmaal onsraons. I s sn ha α q (5.7) Θ F (5.8) hrfor, h spa-m boy fram of h parl roas wh four-nsor angular vloy q Ω F ( ~ x ) (5.9) m rlav o h nral fram. I s sn ha h hyprbol an rular angular vlos of h boy fram ar an η q E( ~ x ) (5.) m ω q B( ~ x ) (5.) m rspvly. Now w ralz ha h lromagn srngh fl nsor an Lornz for vor ar boh a naural onsqun of h gomr sruur of rlav spa m. Bas on our xprn wh onnuum mhans, as w mnon bfor, h srngh nsor F fl sms lk a four-mnsonal vory fl. hs lromagn vory four-nsor fl s a ombnaon of hyprbol lromagn vory E an rular lromagn vory B. I s sn ha h salar m q maps h vory fl F a h poson of h parl o h four-nsor angular vloy Ω of s boy fram. hrfor, h ff of lromagn nraon on a harg parl s nohng bu h nsananous four-mnsonal roaon of s boy fram. h quaons (3.) an (3.) for h parl an b wrn as u ω u + τ v / η (5.) 46

47 η u τ v / (5.3) hs quaons ar quvaln o h spa an m omponns of quaon (5.6) for lromagn nraon as u m mv q v / ( E + v B) (5.4) m v / qe v (5.5) s w know, h frs quaon s h quaon of moon, whr s rgh han s s h famlar Lornz for. h son quaon s h ra a whh h lromagn fl os work on h parl an hangs s nrgy. In ovaran lromagn hory, h four-vor lr urrn nsy J ( J, J 4 ) ( J, ρ ) ρ ( v, ) (5.6) E E E E E sasfyng h onnuy quaon ρe J E, J E + (5.7) gnras h lromagn four-vor ponal, whr E (, 4 ) (5.8) n spa-m orrsponng o h nral rfrn fram. h spa omponn s h magn vor ponal an h m omponn 4 s rla o h lr salar ponal φ as 4 φ (5.9) h four-mnsonal url of gvs h lromagn fl srngh nsor F F (5.) 47

48 hrfor, h fls E an B ar xprss n rms of hs ponals as E φ (5.) B (5.) I shoul b no ha h four-vor V orrspons o h ngav of V (5.3) an an b onsr as an lromagn vloy fl nu n four-mnsonal spa-m rlav o h nral fram. s was mnon prvously, s fourmnsonal url s h lromagn vory four-nsor F Θ F (5.4) h ovaran form of h govrnng quaon for srngh or vory nsor F u o h lr urrn nsy s 4πK F J E (5.5) whh s h ompa form of Maxwll s nhomognous quaons E 4πKρ E (5.6) E 4πK B + J E (5.7) Equaon (5.6) s Gauss s law an quaon (5.7) s mpr s law wh Maxwll s orron. In hs quaons, h onsan K s h lrosa or Coulomb onsan ha usually s wrn as also h rlaon K, whr ε s h prmvy of fr spa. hr s 4πε, whr onsan s all h prmably of fr spa ε 4π K an h rlaon hols. hrfor, h quaon (5.5) an b wrn as F J (5.8) E 48

49 an also h Gauss an mpr s laws (4.6) an (4.7) bom ε E ρ E (5.9) E B + J E (5.3) h ompably quaon for F s σ F + F + F (5.3) σ σ hs s h nssary onon o oban h lromagn vloy from vory fl F. I smply hks f a gvn lromagn vory fl s apabl or no. hs quaon s h ovaran form of Maxwll s homognous quaons B (5.3) B E + (5.33) s w know, h quaon (5.3) s Gauss s law for magnsm an h quaon (5.33) s Faraay s law of nuon. h s of quaons (5.9)-(5.3) an (5.3)-(5.33) ar Maxwll s quaons n SI uns. hy smply show h rlaons govrnng h lromagn vory nu o spa-m. I s sn ha h gomral hory of lromagn nraon s vry lar n SI uns. Inrsngly, s ralz ha h lromagn hory woul hav bn muh mor ompabl wh h gomral hory f h salar an vor ponals φ an, an magn fl B ha bn fn as h ngav of hr prsn form. h four-vor ponal fl s no unquly rmn from ompabl srngh four-nsor F u o h gaug from. In, h nw fl + λ (5.34) 49

50 os no hang h fl srngh nsor F. Suh ransformaon s all a gaug ransformaon n whh h funon λ s a funon of oorna x. hs gaug from allows us o hav h Lornz gaug onsran φ + (5.35) hrfor, λ s no ha arbrary. I mus sasfy h wav quaon λ λ λ (5.36) hs wav quaon an b onsr as rprsnng h nral lromagn wavs. Usng h Lornz gaug n (5.8) prous h manfsly ovaran wav quaon J (5.37) α α E Wha w hav shown s ha Maxwll s quaons ar quaons govrnng h hyprbol an rular angular lromagn vors σ E an B. h quaon F + F + F (5.3) s nohng bu a knma ompably for hs lromagn vors. h nonhomognous quaon σ E σ F J (5.8) s h rlaon among hs vors an lr four-vor nsy urrn. n analogy wh onnuum mhans suggss hs rlaon s h quaon of moon for lromagn vors. Maxwll s quaons ar ovaran, whh mans hy ar nvaran unr Lornz ransformaons among nral sysms. hrfor, h four-vor, an four-nsor F ar funamnal fls npnn of any spf spa-m nu n h unvrsal ny mnon bfor. I s h nral obsrvr who spfs a spa-m n hs unvrsal ny an masurs omponns for hs four-vor an four-nsor, for 5

51 xampl F for F. h omponns of hs four-nsor ransform unr Lornz ransformaon among nral sysms as F Λ Λ F (5.38) α β αβ h non-eulan harar of lromagn fl nsors s obvous from hs ransformaons. Inrsngly, h salars F F ( B E ) (5.39) F ( E B) (5.4) ar h nvarans of h four-nsor F unr h Lornz ransformaons. hy show ha h salar B + E s nvaran. I s obvous ha h non-nral obsrvrs ar no qualf o us (5.38), baus h ransformaon x Λ x (.6) os no hol among hm. W monsra hs fa by a smpl xampl. Consr h lromagn vory fl gnra by a fr harg parl. Is boy fram s an nral fram an has a unform moon rlav o ohr nral obsrvrs. h parl gnras h lr fl n s nral boy fram, suh ha hrfor, s sn ha whr q E 4πε r rˆ (5.4) (,,, φ ) (5.4) q φ (5.43) 4πε r 5

52 s paralll o h four-vor vloy u (,,, ). Rlav o h rfrn nral fram, w hav an x Λ x (5.44) Λ (5.45) I sms as f h four-vor fl wr aah o h boy fram rgly n h m ron, suh ha looks roa rlav o h fx nral sysm. Howvr, hs rg lk harar an ovaran rlaons ar no val whn h parl s alrang. I sms h spa-m boy fram of an alrang parl os no look rg n hs sns o any obsrvr. hrfor, h poson four-vor x, four-vor ponal an four-nsor F fl o no ransform unr a unform hyprbol roaon. hs non-rg harar an b onsr as h gomral orgn of lromagn raaon. h raaon of an alrang parl an b analyz by usng h gnral quaon J (5.37) α α E n h onx of Lénar-Whr ponal []. Wha s h onsqun of h non-rgy of h spa-m boy fram of an alrang parl? s w saw, h four-alraon of a harg parl u q F ~ ( x ) u (5.46) τ m boy fram n h nghborhoo of h parl. hrfor, w fn ( ) s h rsul of rg-lk nsananous roaon of s boy fram. Howvr, w now ralz ha h global rg-lk harar of h boy fram s no a rqurmn for hs gomral rvaon. I s only nssary o onsr h nsananous roaon of h Ω x~ as h fourmnsonal angular vloy of h boy fram a h poson of h parl q Ω ( ~ x ) F ( ~ x ) (5.47) m 5

53 Inrsngly, wh hs vlopmn w an xamn h harar of parls n quanum hory o xplan h wav-parl ualy of mar. I s n lassal mhans whr w spfy poson of a parl, for xampl, a h orgn of s spa boy fram. In quanum mhans, a fr lmnary parl wh spf momnum os no hav a spf poson n s spa-m an an b anywhr n s boy or nral rfrn fram. On an suggs ha h wav funon of h parl rprsns h ra of s spa-m boy fram on h nral rfrn fram. hrfor, s nssary o unrsan h Dra spnor wav funon n h framwork of h prsn spa-m hory. hs nw vw looks vry promsng f w rmmbr ha h wav funon of an nrang parl s loalz an s ffrn from h wav funon of a fr parl. I s sn ha hs s nohng bu h manfsaon of a formaon-lk harar of h spa-m boy fram of an nrang parl. Inrsngly, w ralz ha h raon an annhlaon of parls an b xplan as h rsul of onsrans n h m ron. I s lar ha w may xp o rsolv ambgus n h quanum worl an ohr branhs of morn physs wh our nw vw of spa-m. Hr, w shoul mnon ha h affny of h Lornz ransformaon wh lromagn srngh fl nsor an Lornz for has bn ralz bfor. For xampl, Burago has sa ha h lromagn srngh fl nsor an Lornz for ar boh a naural onsqun of h gomr sruur of Mnkowskan spa m, whh nas a funamnal manng n physs []. Obvously, wha w hav hr s vlopmn of hs funamnal manng. Now s m o xplor mor abou h unvrsal funamnal ny n whh parls ra hr spa-m an nra hrough vory fls. I urns ou ha h rvw of lromagn nrgy-momnum nsor an Maxwll srss nsor s usful. 5.. Elromagn nrgy-momnum nsor Rlav o h spa-m nral rfrn fram, h Lornz for pr un volum on a mum wh a harg nsy ρ E an urrn nsy J E s gvn by f ρ E + J B (5.48) E E 53

54 h gnralzaon of hs for n ovaran lroynams s whr ( ) f F J (5.49) E f f, f 4 s h for-nsy four vor wh f 4 J E E (5.5) W no ha w J E E (5.5) s h work on pr un m pr un volum by h lr fl on movng hargs. hrfor w f4 (5.5) By subsung J E from h quaons of moon of h lromagn fl F J (5.3) E an som nsor algbra, w oban f (5.53) whr s h lromagn nrgy-momnum nsor fn by Fσ Fσ + δ Fαβ Fαβ 4 (5.54) h xpl form of h omponns of hs four-nsor n rms of E an B ar all h Maxwll srss nsor, an j ε ( EE j Ek Ekδ j ) + ( B B j Bk Bkδ j ) (5.55) 54

55 h lromagn nrgy nsy, an whr h Poynng vor S s fn by 4 44 u ε + E B (5.56) 4 B S E B ( E ) S (5.57) (5.58) hrfor, h symmr four-nsor an b wrn n shma marx form as j S (5.59) S u h raon or for xr by hs fl on a un ara of a surfa n spa wh un normal vor n s j j ( n) n (5.6) hrough hs smlary wh onnuum mhans, w an ak as a four-srss nsor. h m-spa omponns of h quaon (5.53) ar f j x j S (5.6) S j u f4 (5.6) x Ingrang hs rlaons ovr a volum V boun by surfa, an usng h vrgn horm, w oban fv + SV V V j n j j (5.63) V f4 V + + uv Sn V (5.64) 55

56 hs quaons show ha h lromagn fl has nrgy an arrs momnum. h Poynng vor S rprsns h nrgy pr un m, pr un ara, ranspor by h fls n spa. I s also sn ha h lromagn fl arrs momnum, suh ha G S ε E B (5.65) s h lromagn momnum nsy vor. By nong an W F f V oal for ang on volum V (5.66) V f4 V work on pr un m by h lr (5.67) V fl on movng hargs n V w oban h quaons (5.63) an (5.64) as ( n F G V ) + (5.68) V W + V uv + S n (5.69) I s sn ha by onsrng P mh F (5.7) whr an an P mh mhanal momnum of hargs n volum V (5.7) P fl GV lromagn momnum n volum V (5.7) V U fl uv lromagn nrgy n volum V (5.73) V w oban h momnum an nrgy onsrvaon laws 56

57 ( n ( P + P ) ) fl mh (5.74) ( W + U ) + S n fl (5.75) hs rlaons n voral form ar ( n ( P + P ) ) mh fl (5.76) ( W + U ) + n fl S (5.77) In aon, no ha h rlaon Fσ Fσ + δ Fαβ Fαβ 4 (5.54) looks lk a onsuv rlaon for four-srss nsor n rm of h four-nsor lromagn vory F n h unvrsal ny. In lnar onnuum mhans, h onsuv quaons rla h srss nsor lnarly o sran or sran ra, bu h nrgy nsy s a quara funon of sran or sran ra nsor. Howvr, wha w hav hr s four-mnsonal analogous as n whh h srss four-nsor s a quara funon of vory four-nsor n h unvrsal ny. hrfor, s sn ha h unvrsal ny bhavs lk a onnuum n whh harg parls ra srsss an lromagn vors. Inrsngly, h pon harg parls ar sngulars of hs vors an four srss nsors. hrfor, h Mnkowsk fors xr on hs pon parls ar h Lornz fors, whh an b onsr also as four-mnsonal lf fors. lhough hs onluson looks vry nrsng, hsoral aouns show s no omplly nw. hs vlopmn s smlar o h ffors of nvsgaors of hr hory. Ehr was h rm us o srb a mum for h propagaon of lromagn wavs. For xampl, s vry nrsng o no ha MCullaugh [] onsr hr o b a nw kn of mum n whh h nrgy nsy pns only on h roaon of h volum lmn of hr. h work of MCullough has bn a bas for work of ohr proponns of hr hory suh as Lor Klvn, Maxwll, Krhhoff, 57

58 Lornz an Larmor. Whakr [3] gvs a al aoun of hs nvsgaons n whh w larn ha Maxwll agr o a roaonal harar for magn fl an a ranslaonal harar for lr fl. W also larn ha Larmor [4] onsr ha h hr was spara from mar an ha parls, suh as lrons, srv as sour of vors n hr. Wha s surprsng s ha w hav us smlar as abou srss an vory, bu n a four-mnsonal onx. In our vlopmn, h magn fl has h sam harar as rular roaon, bu h lr fl has h harar of hyprbol roaon. I s sn ha s wll jusf o all our funamnal unvrsal ny h hsoral hr ou of rsp, whh now s rprsn by four-mnsonal spa-m sysms. hrfor, n h nw vw, parls spfy hr spa-m boy frams n h hr an nra wh ah ohr hrough four-vory an four-srss ha hy ra n h hr. s w mnon, h Lornz for F qf ~ x u (5.4) ( ) s analogous o h lf for n flu ynams. h lf on an arfol s prpnular o h vloy of flow pas h surfa. hs s h mhanal xplanaon of four-vor lromagn Lornz for. I s obvous ha unrsanng mor abou hr an spa-m s an mporan sp owar unrsanng mor abou morn physs. Howvr, h gomral hory of lromagn nraon rsolvs som ffuls vn n hs lassal sa. W arss wo mporan ass. 5.. Magn monopol os no xs Wh h nw vw, h magn fl B s h spa lromagn vory nu o h hr rlav o h rfrn nral fram. hs s analogous o h vory fl n a roaonal flu flow. From non-rlavs flu mhans, w know ha h vory s h url of h vloy fl of h flu an s w h angular vloy of 58

59 h flu lmn. hrfor, w s h sam for h lromagn vory. h magn fl B s h url of h lromagn vloy vor fl B (5.) hs fnon rqurs B (5.3) whh s h knmaal ompably quaon. hs s h nssary onon for h xsn of vor ponal for a gvn magn fl B. Exsn of a magn monopol volas hs rval knmaal ompably quaon. W monsra hs furhr by onraon as follows. L us assum, a h orgn, hr s a pon magn monopol of srngh q m. hrfor, n SI uns ( 3) B q δ ( ) (5.78) m x an h sa magn fl s hn gvn by q B m rˆ (5.79) 4π r Howvr, h rlaon (5.78) onras h knmaal ompably (5.3). Inrsngly, bas on h Hlmholz omposon horm, hs fl an only b rprsn by a salar ponal [5] q φ m m ( x ) (5.8) 4π r whr h magn fl B s gvn by B φ m (5.8) Bu hs s absur baus h lromagn vory vor fl B has o b always rprsn by url of h lromagn vloy vor. hrfor, magn monopols anno xs. I s onlu ha h magn fl B s only gnra by movng lr hargs. 59

60 I has bn long spula ha magn monopols mgh no xs baus hr s no ompl symmry bwn B an E. hs s u o h fa ha B s a psuo-vor, bu E s a polar vor. Wha w hav hr s h onfrmaon of hs orr spulaon ha hr s no ualy bwn E an B n lroynams. W hav shown ha h magn fl B has h harar of a rular vory fl an s vrgn fr. Howvr, h lr fl E has h harar of a hyprbol vory wh lr hargs as s sours, whr ε E ρ E (5.9) I s sn ha hs xplanaon s aually larfaon of Larmor s hr hory. s mnon prvously, h lr harg q of a parl has h propry of a knmaal ouplng, whh maps h four-mnsonal lromagn vory a h poson of h parl o h angular vloy of s boy fram. W hav shown ha lr harg s h only ouplng prsn. Furhrmor, hr s no n for any ohr ouplng. I s naïv o assum ha a smpls mofaon of Maxwll s quaons suff o allow h xsn of magn hargs n lroynams Spn ynams an magn momn I s known ha vry lmnary parl, suh as an lron, has an nrns angular momnum all spn. h spn an b onsr as a onsan lngh four-vor ( ) s s, s 4 suh ha rlav o h parl boy fram, h spn four-vor has only spa omponns. hs mans ha s normal o h parl s four-vor vloy rlav o s fram an also h nral rfrn fram u s u s (5.8) If h lromagn fls ar unform, h quaon for spn s gvn by h BM quaon ) 6

61 s q g g F s + us Fλuλ τ (5.83) m whr g s all h gyro-magn rao. By usng an analogy wh orbal angular momnum of sysms of harg parls an h onp of magn pol momn, w an show g. Howvr, xprmns show s a numbr vry nar. h Dra rlavs wav quaon for an lron shows g []. hrfor, h BM quaon boms s q F s (5.84) τ m hs s fanas! I s sn ha h valu g s ompabl wh h vlop gomral-knmaal hory of lroynams. h spn four-vor s an aah four-vor, whh s roang wh q Ω F ( ~ x ) (5.9) m hrfor, h onsan lngh spn roas wh h boy fram, suh ha s q Ω s F s (5.85) τ m I shoul b no ha h spn four-vor has only spa omponns n s boy fram, whh s onssn wh (5.8). Inrsngly, now w ralz ha h analogy o orbal angular momnum an usng h onp of magn pol momn, whh las o g, s mslang. 6. Maxwllan hory of gravy h Maxwllan hory of gravy gnralzs h Nwonan hory of gravy o movng masss. I s lar ha hs s h ompabl hory wh our gomral hory of nraon. h pulary of hs hory, alhough lassal hory offrs no ompllng 6

62 rason bhn, s ha h gravaonal harg m G s proporonal o h nral mass m, as far as w know. hs s all h quvaln prnpl, whh mans n a propr sysm of uns, suh as h SI sysm, hs wo masss ar qual m G m (6.) Howvr, shoul b no ha n h vlop gomral nraon hory, h quvaln prnpl s no a funamnal nssy a all. If, n fuur, hs prnpl s nvala n som rang of masss, hs hory wll sll rman val. In hs hory, h gravaonal mass (harg) nus h four-momnum pr un gravaonal mass or gravaonal four-vloy U, whr U U ( U, U 4) (6.) o h hr rlav o h spa-m nral obsrvr. Baus of h quvaln prnpl, h gravaonal four-vloy fl U looks lk h four-vloy u of h parl. hs xplans why w us h symbol U o rprsn hs vloy-lk fl. By analogy o h lromagn hory, U 4 shoul b rla o h salar Nwonan ponal Φ. I wll b shorly shown ha Φ U 4 (6.3) h an-symmr four-nsor gravaonal nnsy fl s hararz by h url Ω G ( U U ) (6.4) whh s h gravaonal four-vory nu n h hr masur by an nral obsrvr analogous o F n lroynams. W hav hosn h symbol Ω o mphasz h analogy of h spa gravaonal vory o vory n lassal flu mhans. In rms of omponns G 6