THE CARTAN GEOMETRY OF THE PLANE POLAR COORDINATES: ROTATIONAL DYNAMICS IN TERMS OF THE CARTAN SPIN CONNECTION

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1 Jounl of Foundtions of Physics nd Chmisty 3 HE CARAN GEOMERY OF HE PLANE POLAR COORDINAES: ROAIONAL DYNAMICS IN ERMS OF HE CARAN SPIN CONNECION M. W. Evns nd H. Eck Alph Institut fo Advncd Studis ( Ctn gomty is pplid to th pln pol coodints to clcult th ttd nd spin connction lmnts fom fist pincipls of gomty. It is shown tht th Ctn tosion is non zo fo th pln pol coodints, thus futing Einstinin gnl ltivity. h ltt ssums incoctly tht th tosion is zo. Simpl clcultions bsd on th pln pol coodints show tht th Ctn tosion is spcil cs of mo gnlly dfind tosion, spcil cs in which th connctions qul nd opposit in sign. hs nw mthmticl tchniqus pplid to ottionl dynmics, nd it is shown tht th ngul vlocity is Ctn spin connction. Kywods: ECE thoy, Ctn gomty of th pln pol coodints, Ctn gomty of ottionl dynmics.. INRODUCION In this sis of pps nd books [-] th ECE gnlly covint unifid fild thoy hs bn dvlopd on th bsis of Ctn s wll known gomty [] in which th two stuctu qutions usd to dfin tosion nd cuvtu. It hs bn shown tht Einstinin gnl ltivity (EGR) is incoct bcus of its nglct of on of th fundmntls of gomty, th Ctn tosion. In Sction th Ctn tosion is clcultd with th pln pol coodints, nd shown to b non-zo. his simpl xcis futs EGR bcus in ny gomty, th Ctn tosion is in gnl non-zo. It ws shown in th fist pps of ECE thoy tht th Ctn ttds cn b dfind by using ny two coodint systms in ny mthmticl spc in ny dimnsion. h oiginl concpt by Ctn [-] usd tngnt spctim t point P to bs mnifold. h tngnt spctim

2 M. W. Evns nd H. Eck in Ctn gomty is Minkowski spctim if fou dimnsionl thoy is bing usd. Diffnt typs of tngnt spctim cn b usd. By supimposing on coodint systm on noth in th sm mthmticl spc, ttds cn b dfind th most simply. his is don in Sction by using points in th pln pol coodint systm nd points in th Ctsin systm. h nlysis is ducd to th simplst possibl lvl by considing pln. A vcto cn b psntd by th pln pol coodints [-4]. h ttd lmnts th Ctsin componnts of th vcto in pln pol psnttion. h Ctn spin connction is dfind by th fct tht th xs of th pln pol systm ott with spct to th fixd xs of th Ctsin systm. Hving dfind th ttd nd spin connction componnts th fist nd scond Ctn stuctu qutions usd to clcult th Ctn tosion nd cuvtu of th pln pol coodints in two dimnsions. h most impotnt sult is obtind tht th tosion is not zo. If th Ctn tosion is not zo on th simplst possibl lvl, thn it is not zo in ny gomty. his splls disst fo stndd physics bcus EGR is bsd on zo tosion. his tchniqu is usd in Sction 3 to show tht th ngul vlocity is Ctn spin connction. h ltt is thfo fundmntl to ll th fmili concpts of ottionl dynmics. As usul th nots tht ccompny this pp on giv lot of dtil of th clcultions, nd should b d in conjunction with this pp, UF35.. CALCULAION OF HE CARAN ORSION Consid th wll known [-4] unit vctos of th pln pol coodints: cos i + sin j sini+ cos j h unit vctos dpnd on tim [] nd ott. h unit vctos i nd j of th Ctsin systm do not dpnd on tim nd fixd o sttic. h fou lmnts of th Ctn ttd q dfind by: µ q q i q q j (3) his is n xmpl of th gnl dfinition []: h fou ttd componnts : nd th ttd mtix is: q q V µ q µ V cos, q sin, q µ sin, q cos, cos sin sin cos (4) (5) (6)

3 h Ctn Gomty of th Pln Pol Coodints 3 Not cfully tht this is lso th ottion mtix bout Z: V X VX V cos sin Vy y sin cos fo ny vcto V. It follows fom Eqs. (4) nd (7) tht: wh: o pov this sult not tht: so: V V V cos sin sin cos V V V, V V, X V V, V V X y V V i+ V j V + V X y ( ) V cos i+sin j + V sin i+cos j V i+ V j y (7) (8) (9) () V V cos V sin V V sin+ V cos Multiply Eq. by cos nd Eq. by sin. It follows tht: ( V ) V cos+ V sin, (3) ( V ) V sin+ V cos (4) which is Eq. (8), QED. It hs bn povn tht ottion in th pln XY bout Z dfins th Ctn ttd mtix nd fou lmnts of th Ctn ttd. Dfin th mtic in th Ctsin systm by g µ~ nd th mtic in th pln pol systm by η b. By dfinition [] th two mtics ltd by: h mtics ltd to th infinitsiml lin lmnt by: g q q b µ ~ µ ~ ηb (5) ds µ g µ ~ dxdx ~, (6) so [ -4]: ds b η dx dx, (7) b ds dx + dy d + d, (8)

4 4 In Ctsin coodints: M. W. Evns nd H. Eck ds gdxdx + g dx dx (9) nd in pln pol coodints: dx dx, dx dy, g g, () Eq. (5) mns: dx d, dx d, g g g q q η + q q η, g q q η + q q η (3) Fom Eqs. (5), () nd, Eqs. nd (3) coct nd slf sufficint. Eqs. nd (3) both giv: Fom ottion gnto thoy [4], if: R Z cos + sin (4) cos ( ) sin sin cos thn th opto known s th ottion gnto is dfind s: In th dimnsions: nd: It follows tht [4]: (5) dr i Z J Z (6) i d i i JZ i, JX i, J i o pov this consid th Mcluin sis: R Jx, J y ij z t cyclicum z Y i i (7) (8) ( ) xp ( ij ) (9) z

5 h Ctn Gomty of th Pln Pol Coodints 5 xp( ij z ) + ij z J z +...! cos sin sin cos + +! hfo th ttd mtix is: (3) qµ xp ij z (3) wh both sids mtics in this nottion. h gnl stuctu of Eq. (6) is tht of th divtiv of ttd, bcus th ottion mtix is ttd mtix. h ttd postult of Ctn gomty ssts [] tht th covint divtiv of th ttd is zo: his qution cn b xpssd s: b λ Dq q + q Γ q (3) ~ µ ~ µ ~ b µ ~ µ λ q ζ : Γ ( ) (33) ~ µ ~ µ ~ µ ~ µ nd is gnliztion of Eq. (6). hfo th ottion gnto is spcil cs of th zt connction dfind by: ζ : Γ (34) ~ µ ~ µ ~ µ h Ctn tosion ssocitd with this pocss is dfind to b [-] b b q q + q q Γ Γ (35) µ ~ µ ~ ~ µ µ b ~ ~b µ µ ~ ~ µ Any obit in pln is gntd by connction nd tosion of gomty. his is nw undstnding of ll obits in pln nd oigints in th fct tht fo ny such obit: It follows tht: so fom Eq. (6): d d (36) dcos d sin, (37) d d dsin d cos, (38) d d dqµ d sin cos d cos sin d Fom Eq. (33) th zt connction mtix of ny pln obit is: ζ µ d sin cos cos sin d (39) (4)

6 6 M. W. Evns nd H. Eck Any pln obit is popotionl to th zt connction, which is ottion gnto popotionl to ngul momntum nd ltd to spctim tosion. Fo xmpl, fo th llipticl obit [-]: nd th zt connction lmnts : α ζ ζ, α ζ ζ cotn d sin (4) d α h unit vctos of th pln pol systm ott nd th Ctn spin connction dfins th ottion. Dnot th bsis vctos by: By dfinition [] th covint divtiv is dfind by (4), (43) µ µ µ ( b) b D + (44) Howv th odiny fou divtiv vnishs bcus it is dfind with sttic fm of fnc. h spin connction is dfind to dscib th otting fm. Fo xmpl: nd hfo: In vcto nottion: D D d ( ) ( b) + ( b) (45) D ( ) d (46) b + (47) ( b) (48) his sult is lwys dnotd [-4] by d d (49) but igoously it should b: D t d + t (5) sttic

7 h Ctn Gomty of th Pln Pol Coodints 7 It follows tht (5) sult of bsic impotnc. With th dfinitions [-4] Eq. (5) cn b xpssd s: k, (5) k, (53) k, (54) D t d +, (55) sult of bsic impotnc to ottionl dynmics (Sction 3). Similly th bsic kinmtic sult []: d (56) cn b xpssd s: D sttic So th two spin connction lmnts of this typ : (57) (58) All th infomtion ndd to clcult th lmnts of th Ctn tosion is now vilbl. Fo xmpl: b b q q + q q (59) ( b) b with summtion ov ptd indics (b). By dfinition: so th sult ducs to: q, q q, (6) q + q (6) Using th lvnt ttd nd spin connction lmnts givs th sult: d sin sin (6)

8 8 M. W. Evns nd H. Eck Pocding similly givs th tosion mtix: which cn b xpssd s: hfo: ~ µ i sin cos d d q d cos sin ~ d sin cos µ µ cos sin (63) (64) i J i z (65) wh th infinitsiml ottion gnto bout Z is dfind [4] s: J z i h finl sult is tht th tosion mtix is popotionl to th infinitsiml ottion gnto: i J z (67) Fom th ttd postult (3): i ( ) ( ) b Γ µ ~ µ q ~ µ ( q b) ~ so th xist mixd indx connctions such s: Γ (66) + (68) ( q + q sin (69) Fom Eq. (59): Γ sin, Γ, (7) so ths mixd indx connctions nith symmtic no ntisymmtic fo th pln pol coodints, fo which th Ctn tosion is in gnl non zo. his sult is lon nough to fut ny thoy of ltivity bsd on zo tosion, notbly EGR. h mixd indx connction of typ (68) cn lwys b dfind s th sum of symmtic (S) nd ntisymmtic (A) componnts by bsic thom of mtics [5]. h Ctn tosion lmnt (6) is dfind [-] to b: Γ A Γ A (7) nd is lwys twic th ntisymmtic connction Γ ( A ). So: Γ ( A) (7)

9 h Ctn Gomty of th Pln Pol Coodints 9 Mo gnlly [] ny diffnc of connctions such s: µ ~ µ ~ ~ µ Γ Λ (73) is tosion tnso. Clly, th Ctn tosion (7) is spcil cs of th gnl dfinition in Eq. (73) []. hs lmnts of th tosion my b givn th pplltion obitl tosion [-] in nlogy with obitl ngul momntum. h lso xist spin tosion lmnts such s: b b q q + q q (74) b b h vlution of th spin tosion fo th pln pol coodints quis th vlution of diffnt spin connctions fom thos usd in obitl tosion. Fom th bsic dfinitions of th unit vctos of th pln pol coodints: it is sn tht: i cos+ j sin, (75) isin + jcos, (76) d d h infinitsiml lin lmnt is dfind by: d d,, (77) d d so Dfining: µ ~ ds g dx dx g dx dx + g dx dx µ ~ d + d x, x,, (78) (79) y (8) thn: Eq. (77) my b wittn s: so: d dx df dy df d d d d, d d (8) d d d, i.., (8) d d d d (83)

10 Eq. (8) my b wittn s: so M. W. Evns nd H. Eck d dx (84) d (85) d hfo th lvnt spin connction componnts d d It follows tht: in which: So: Similly: dsin dcos + q + d d ( ) q q so th spin tosion lmnt is ntisymmtic: q (86) (87) (88) d ( cos+ sin ) (89) d d ( cos+ sin ) (9) d s it must b by dfinition [-], QED. In od to vlut th spin tosion lmnt: (9) (9) nw spin connction lmnts hv to b dfind. Fom Eqs. (75) nd (76): d d (93) d d nd: d d d( ) d (94) Eq. (93) is: d dx (95)

11 nd Eq. (94) is: h Ctn Gomty of th Pln Pol Coodints d dx (96) so th quid lmnts of th spin connction : hfo: d d (97) d ( cos sin ) (98) d h lvnt mixd indx connctions : Γ d cos Γ, (99) d giving th mtix: Γ Γ Γ d sin Γ, () d Γ Γ d d cos sin cos sin () hfo th spin tosion lmnts of th pln pol coodints non-zo, sult of bsic impotnc tht gin futs EGR. In gnl th mixd indx connctions of typ () nith symmtic no ntisymmtic. Using th dfinition (7) th lvnt ntisymmtic connctions : Γ d + A ( cos sin ), () d In vcto fomt, using Γ ( ) d A ( cos sin ) (3) d 3 (4) 3 th spin tosion is dictly popotionl to th vcto ngul vlocity: cos + sin d d k (5) In ltivistic contxt th units of tosion invs mts, but in this non ltivistic contxt th obitl tosion is dfind to hv th units of invs sconds.

12 M. W. Evns nd H. Eck 3. SPIN CONNECION AND ANGULAR VELOCIY IN ROAIONAL DYNAMICS h sult (55) of Sction fo unit vctos is tu fo ny vcto V, bcus ny vcto cn b xpssd in tms of unit vctos. hfo: DV dv + V (6) his is th fmili sult of clssicl ottionl dynmics []. In pln pol coodints [] th position vcto fo xmpl is: so: V d d + d (7) V + (8) wh v is th totl vlocity. It is th sum of th vlocity in n intil fm (fm with sttic coodints): d Vs (9) nd th obitl lin vlocity []: In componnt fomt: s V () d V + + W iv t th impotnt sult tht th obitl lin vlocity is th sult of th Ctn spin connction in pln pol coodints. Fom Eq. th ccltion is dfind by: dv d d d + d d d d + + h tm in bckts is th ccltion du to th ottion of th fm of fnc itslf, so is du to th Ctn spin connction. Fom fundmntls []: so: d d / (3) d d + + d ( ) (4)

13 In this qution: h Ctn Gomty of th Pln Pol Coodints 3 nd th intil o Nwtonin vlocity is: hfo: Using: th ccltion is: in which, d d d ( ) + (5) v d N (6) dvn d + vn + + (7) d d (8) dv N d + vn + (9) nd: k () h complt ccltion is thfo: nd in componnt fomt is: ( ) k d d + vn + + ( ) + ( + ) (3) h Coiolis ccltion is: d co vn + ( + ) (4) nd th cntifugl ccltion is: h intil o Nwtonin ccltion is: cnt ( ) (5) d Nwton (6) W iv t th impotnt sult tht ll ths wll known ccltions du to th Ctn spin connction, which in vcto fomt is th ngul vlocity.

14 4 M. W. Evns nd H. Eck In pvious wok [-] it ws found tht th Coiolis ccltion vnishs fo ll pln obits: d co vn + (7) so th totl ccltion fo ll pln obits is: d (obit), + (8) nd is th sum of th intil nd cntifugl ccltions. Fom pvious wok it ws found tht th intil ccltion of th llipticl obit is: d L m wh L is th consvd totl ngul momntum: h llipticl obit is dfind by: L α (9) m (3) + α cos wh α is th hlf ight ltitud nd є th ccnticity. Fo th cicul obit: (3) α (3) so th intil ccltion of th cicul obit vnishs: h intil foc is dfind to b: d F m d N nd this is gnl sult vlid fo ll pln obits. In th pticul cs of Nwtonin dynmics []: (33) (34) F N L m 3 L L, α (35) m α mmg so th intil foc is: F N L mmg m (36) 3 In th civd opinion [] this sult is intptd s th foc of ttction of th invs squ lw:

15 h Ctn Gomty of th Pln Pol Coodints 5 ddd to psudo-foc dfind by: Ftt mmg µ c F L psudo 3 m (37) (38) his psudo foc is dfind incoctly s oiginting in n ffctiv potntil. Howv, th complt ccltion is: wh L L 3 + ( ) (39) m mα L ( ) (4) m so th coct sum of ccltions in Eq. (39) consists of only on tm: ( sum) L m α (4) nd this is igoously coct sult tht oigints in th bsic dfinition of ccltion. Using Eq. (3) fo th ngul momntum it is found tht th totl ccltion ssocitd with th llipticl obit is: ( ) α α (4) nd is du ntily to th spin connction, i.. to th ottion of th xs nd of spc itslf. W iv t th impotnt conclusion tht vy pln obit is du to th movmnt of spc itslf. Fo cicul obit: so: A solution of this qution is: th l pt of which is: (43) d (44) ( xp ) ( i t) (45) Rl ( cost ) (46)

16 6 M. W. Evns nd H. Eck so th vcto otts in cicl with ngul vlocity, which is lso th mgnitud of th spin connction of Ctn. Fo n llipticl obit: d α (47) W iv t th impotnt conclusion tht th Nwtonin intpttion is untnbl bcus th coct totl ccltion (4) is intptd s foc of ttction. It is sn fom Eq. (39) fo xmpl tht th Nwtonin pocdu dds nd subtcts tm: L m 3 + (48) fom th coct sult (43). h Nwtonin thoy ws dvisd fo th llipticl obit, in which cs it givs th sult (47). his is not foc of ttction, it is th qution of otting vcto, Eq. (4). h pln obit is not du to blnc of ttction nd psudo foc, it is du ntily to th motion of spctim itslf nd to th Ctn spin connction. h totl ccltion of th llipticl obit is lwys: L + ( ) m α (49) which is spcil cs of th gnl sult fo ny pln obit: d + (5) Whn th ngul vlocity vnishs, th ccltion ducs to th intil: d MG (5) in which cs n objct of mss m is ttctd to noth objct of mss M without movmnt of th xs of th fm of fnc. h tditionl Nwtonin viwpoint of n obit is: mmg E mv, (5) V ( mmg L ) + m, (53) mmg L F + 3 m, (54) nd oigints in th intil tm. It compltly omits th spin connction. In th Nwtonin viwpoint th hlf ight ltitud is chosn to giv n llips povidd tht L α mmg (55)

17 ACKNOWLEDGMENS h Ctn Gomty of th Pln Pol Coodints 7 h Bitish Govnmnt is thnkd fo Civil List pnsion nd th stff of AIAS nd oths fo mny intsting discussions. Dv Buligh is thnkd fo posting, nd Alx Hill, Simon Cliffod nd Robt Chshi fo tnsltion nd bodcsting. Victo Ricnsky is thnkd fo typstting. h AIAS is dministd by th Nwlnds Fmily ust (st. ) nd is st up s not fo pofit ogniztion in Bois, Idho, U. S. A. REFERENCES [] M. W. Evns, Dfinitiv Rfuttions of th Einstinin Gnl Rltivity (CISP,, www. cisp-publishing.com), spcil issu six of fnc two. [] M. W. Evns, Ed., Jounl of Foundtions of Physics nd Chmisty (CISP onwds, six issus y). [3] M. W. Evns, S. J. Coths, H. Eck nd K. Pndgst, Citicisms of th Einstin Fild Eqution (CISP ). [4] M. W. Evns, H. Eck nd D. W. Lindstom, Gnlly Covint Unifid Fild hoy (Abmis Acdmic 5 to ) in svn volums. [5] L. Flk, h Evns Equtions of Unifid Fild hoy (Abmis 7). [6] K. Pndgst, h Lif of Myon Evns (Abmis ). [7] M. W. Evns nd L. B. Cowll, Clssicl nd Quntum Elctodynmics nd th B(3) Fild (Wold Scintific, ). [8] M. W. Evns nd S. Kilich, Eds., Modn Nonlin Optics (Wily 99, 993, 997 nd ) in six volums nd two ditions. [9] M. W. Evns nd J.-P. Vigi, h Enigmtic Photon (Kluw, Dodcht, 994 to ) in tn volums softbck nd hdbck. [] M. W. Evns nd A. A. Hsnin, h Photomgnton in Quntum Fild hoy (Wold Scintific 994). [] S. M. Coll, Spctim nd Gomty: n Intoduction to Gnl Rltivity (Addison Wsly, Nw Yok, 4). [] J. B. Mion nd S.. honton, Clssicl Dynmics of Pticls nd Systms (Hcout, Nw Yok, 988, thid dition). [3] W. G. Milwski (Chif Edito), h Vcto Anlysis Poblm Solv (Rsch nd Eduction Assocition, Nw Yok, 987). [4] L. H. Ryd, Quntum Fild hoy (Cmbidg Univsity Pss, 996, scond dition). [5] G. Stphnson, Mthmticl Mthods fo Scinc Studnts Longmns, London, 968)

Path (space curve) Osculating plane

Path (space curve) Osculating plane Fo th cuilin motion of pticl in spc th fomuls did fo pln cuilin motion still lid. But th my b n infinit numb of nomls fo tngnt dwn to spc cu. Whn th t nd t ' unit ctos mod to sm oigin by kping thi ointtions