Gavy Feld and Eleomane Feld Fne Geomeal Feld Theoy o Mae Moon Pa Two Xao Janha Naal ene Fondaon Reseah Gop, hanha Jaoon Unvesy hanha, P.R.C Absa: Gavy eld heoy and eleomane eld heoy ae well esablshed and onmed by epemens. The hwazshld me and Ke Me o Ensen eld eqaon shows ha he spaal deenal o me ae s he avy eld. Fo pe me dsplaemen eld, when s spaal deenals ae ommave, onsevave elds an be esablshed. When s spaal deenals ae non-ommave, Mawell eleomane eld eqaons an be esablshed. When he ona-ovaan s eqed o he non-ommave eld, boh Loenz ae and Colomb ae ae deved n hs eseah. The pape shows ha he lh s a speal mae n ha he addon o s Newonan mass and s Colomb ele hae s zeo. In a, hs onlson s e o he eleomane wave n vam. Fo he onsevave eld, he eseah shows ha one he mass densy and he Colom hae dendy ae ven, he mao spaeme eae s ompleely deemned. Boh o hem ae nsn eaes o mao mae n osm bakond. Howeve, o he osm aes old evens, he spaal vae may be anno be noed. On hs sense, he oldes avy eld has he laes vae o spae. Ths pon s vey nns o asonomy maes. Key Wods: eneal elavy, eleomane eld, avy eld, moon ansomaon. Inodon In he pape Ineal ysem and peal Relavy [], he symme mae moon ansomaon s dsssed. Fo Newonan mae moon, he vam s aken as a speal mae wh one nns physal paamee-lh speed. Fo any mae, an be dened wh o-movn oodnaos wh loal vayn eomey. The loal vayn eomey epesens he moon o mae n onsdeaon. The moon eqaon o mae s deemned by osm envonmen whee he mae esn, whee wo nns physal paamees ae noded o dene mae as,. When he me aden o me dsplaemen s oed o be zeo, he Newon s mehanal eqaon s oen. I shows ha he Newon s mass s deemned by he shea eae o spae-me onnm. The Newon s mass s epessed as ρ /. When he mae has no mao spaal moon, sh as avy eld o eleon eld, he me dsplaemen eld s noded o desbe he mae moon. One nds ha avy eld, eleo-mane eld, and qanm eld ae me aden eld. They ae elaed wh Newon s mehans n nns sense. When hee s no eleal hae qany, he avy mass and he neal mass n Newon s mehans ae he same. The mae has hee ypal esn oms: aveln wave, loalzed hamon vban
pale, and eponenal epandn o deayn mae eld. The ommable mae moon denes onsevave eld. The non-ommable mae moon denes qanm eld. The wave pale daly s eplaned by deomable mae moon. In hs pa, pe me dsplaemen eld wll be dsssed. In hs ase he moon ansomaon s non-symme. The eseah shows ha he eleomane eld an be dened by he pe me dsplaemen eld, he Mawell eqaons ae deved n hs pa. Fsly, he bas eqaons ae evewed smply. The mae moon n o-dmensonal spaeme onnm s dened by: Tha s he nal bas veos and en bas veos mee ansomaon eqaon: F The mae moon ansomaon enso F s deemned by eqaon: F δ whee, epesens ovaan devave wh espe o oodnao, δ s an n enso. The dsplaemen eld s also dened n he nal o-dmensonal o-movn oodnao sysem.theeoe, he mae moon s epessed by dsplaemen eld meased n sandad physal measn sysem. Is hee dmenssonal ase has been sded by Chen Zhda [-]. Geneally, he mae moon ansomaon enso s non-ommave. Is ovaan nde and ona-vaan nde ae dened espes wh o he nal o-dmensonal o-movn oodnao sysem. The aon o mae moon s dened as: Aon W F d d d d whee, W s a eneal non. The leas aon pnple [-8] ves he eld moon eqaons as: σ 5 whee, k l l σ C F F 6 l k k C k l W 7 k F F l
W F 8 Usally, one does no ae how he onal mae moon sae s n he absole spaeme. One s akes he mae nde dssson as a sel-es obe. In hs sense, he nal mae an be eneally dened by sele he nal o-movn oodnao sysem o make: F δ On hs sense, he eeene mae s aken o dene he spaeme onnm whee he mae nde dssson s movn. The pnple o physal laws ovaan nvaane eqes he eneal eeene mae 9 moon be oodnao ndependen. To mee hese eqemens, he enso k C l ms be soop enso. Theeoe, one has: k C l δ δ δ δ k l l k whee, and ae he nns eae o mae een o eeene mae. Fo Newon mehans, he eeene mae s vam o ehe. Fo he soop enso n me om please ee [9]. In sandad physal heoy, he med om s esed. Howeve, he eseah shows ha he me om s neessay o show he non-ommave eae o mae moon ansomaon enso. Ths op shold ea n e ahe a pesen. Fo smples Newon moon, he elaed physal om o deomaon enso and sess enso ae: F δ The neal mass densy s dened as: σ ρ I nepes he as he nns neal eney o mae. The symme moon ansomaon n he om:
F δ Whee, ae lassal pnple spaal san omponens. The symmey ondon eqes ha:,, 5 Fo smply, he a hee s vewed as physal omponen n a s no. The oespondn physal sess omponens ae: δ σ / / / 6 Fo non-ompessble mae dened by:, he bas mae moon oms ae: mae wave, wave-pale qanm, and onsevave eld avaon eld o sa eleal eld. Alhoh hee ae many pons shold be leaed, o epess he man deas o hs eseah, he pape wll o sah a way. Only when he whole eseah s aelly polshed a lae me, some amby an be eevely emoved.. Mae Moon Who Mao paal Moon Thee ess sh a knd o mae moon ha me dsplaemen aden ae non-zeo whle he spaal dsplaemen aden ae all zeos. Noe ha he physal omponens o me d wll be sed n physal moon eqaon n he sense ha: d d. h a eamen wll make he physal meann o be moe lea he os s only he mahemaal om. The ovaan devave o sess o s smply eplaed by d. Ths s o smply sake, as he ppose o he pape s o show he physal elaon beween vaos physal elds. Fo he mae moon who mao spaal moon, he deomaon o spae-me onnm s:
5 F δ 7 The sess enso s: δ σ 8 Is sess eld s omposed o an soop sess enso and a non-symmeal sess enso. The soop sess enso shows he mae may be a pe onnm mae eld be named as eld mae, hee ae. Ths s snanly deen om Newonan mae n ha he Newonan mae s elaed wh symmeal sess enso. The eqaon 5 shows ha eld mae moon eqaons ae: 9 The ommave and non-ommave ases wll be dsssed bellow.. Consevave Feld Mae Moon Eqaon Commave Case I he paal deenal o me dsplaemen s ommable abo spae and me deenaon, he mae wll behave as onsevave eld mae. As he ommably o spae and me deenaon wll make he eqaon 9 be ewen as: A ρ A ρ A ρ A q A whee: ρ ρ q, q
And ρ A s dened as mae omplee hae qaly; sala A s dened as he poenal o mae eld. I an be seen ha A may epesen avy eld and/o sa eleon eld. Based on hs ndesandn, avy eld and eleon eld ae poded by me dsplaemen aden. Fo avy eld, he hwazshld s solon o avy eld [6-8] nodes Newon s avy eld wh he help o me ae deenaon. The eqaon has hee ypal solons, whh depend on he eae o osm oe n he ase ha s a onsan. Fo he ea physal meann o wll be dsssed lae As depends on he aon non me vaaon, so an be ndesood as eney a pesen. In he pape, s ohly named as osm oe who dealed dssson. a. Absole Tme eloy olon: Gavy Feld and a Ele Feld I,ha s hee s no me aeleaon, he solon s: A B Whee A s a spaal non. In hs ase, he spaal oes o,, ae: A ρ ρ Ths s he onsevave eld n adonal mehans, whee, he poenal o mae eld A s a spaal non. Ths nlds eleon eld and avy eld. Deals abo please see he pape Ineal ysem and peal Relavy []. As ρ, aodn o o pevos eseah n he pa one o hs ll eseah pape, he s Newonan neal mass densy. Theeoe, he anohe pa shold be dened as he ele hae Colomb hae. The ele hae densy may dened as q, o q. Unde above denon, he neal mass and he sa ele hae ae elaed n nns sense as: q ρ, o q ρ 5 Now, s lea ha one he mass densy and he Colom hae dendy ae ven, he mao mae eae s ompleely deemned. Thee o boh o hem ae nsn eaes o mao mae n osm bakond. Fo massless neal mae,, he hd knd o mae onsevave eld an es. Tha s he pe π / hae eld. Is spaal o eld s: A 6 As o ele hae-less mae wll eqe he ondon: q, ha means:. In hs sense, s he neave mae, ha s: 6
A ρ 7 The neave mae has been dsssed n physs o lon me. Hee, he eseah vews he neave mae he mae dened by paamee. I one akes he mae and neave mae as he bas paamee, he eleal hae wll be a devave paamee los s bas poson. As he physs has aken he mass and eleal hae as bas paamee, hee s no need o node he neave mae onep alhoh may be helpl o some ases. b. Consan Tme Aeleaon olon: Tme Dependen paal Foe I >, q >,o <, q <, one wll e he me dsplaemen: C ep[ ] D ep[ ] 8 q q The spaal oes o,, ae: C D ρ ρ { ep[ ] ep[ ]} 9 q q q Ths solon oesponds o bas pale eld n qanm mehans, whee C and D ae spaal non. The spaal oe s me neasn o deeasn. In essenal sense, hee s no seady solon. I onms ha bas pales an be poded o deayed.. Consan Tme Deeleaon olon: Qanm Wave paal Foe I <, q >,o >, q <, one wll e he me dsplaemen: E ep[ ] F ep[ ] q q Whee, he n he h sde o eqaon s he sn o manay nmbe; E and F ae spaal non. The spaal oes o,, ae: E F ρ ρ { ep[ ] ep[ ]} q q q Ths solon oesponds o sel-oaon o pale o vban eld. I shows how he eleon eld and avy eld ae opled oehe n wave om. In a, o posve eney, he solon shows he moon o pales wh neave ele hae. The b and ases show ha he me eae o ommave eld mae moon s deemned by he ele hae and osm oe. In a, hey ae he bas om o molela o aom qanm eld, whee he el hae play an mpoan ole. The eqeny o qanm an be dened as: 7
ν q When he osm oe s physal omponen om, ha s:, he eqeny s: ν q As he bas n o ele hae s a lm qany eqeny an be epessed as: 9 e.689 C, he ν e N whee, N s an nee nmbe wh he same sn wh. I means ha o ed, he eqeny s dsee ahe han onnos. Ths s ased by he dsey o ele hae. On he ohe hand, an be neeed ha o he mae wh vey lae ele hae qany he eqeny wll be onnos. Hene, he qanm phenomeno s elaed wh samll hae mae. Noe ha s onsevave qanm eld.. Non-ommable Feld Mae Moon Eqaon Qanm Mehans I he paal deenal o me dsplaemen s non-ommave abo spae and me deenaon, he mae wll behave as qanm eleomane eld mae. The eqaon 9 be ewen as: A B A B 5 A B A whee, B,,, 6 As A s denes he onsevave mae eld, he B shold be dened as he veo poenal o mae eld, sh as mane eld. I shold be noed ha he oe poded by he veo poenal o mae eld s elaed wh Newon s mae mass. The eqaon 5 has hee ypal solons, whh depend on he eae o osm oe n he ase ha s a onsan. a. Mawell Eleomane Feld Eqaon and Coheen Lh I he spaal oe s zeo, ha s, one wll e: 8
A B A B 7 A B One eonzes ha he eqaon 8 s Mawell s eqaon o eleo-mane eld n vam. Ths eqaon s a bas eqaon o qanm mehans. In veo om, he elaed eqaons ae: D A q 8 H B ρ, H 9 They ae denon eqaons o ele eld E and mane eld H. In hs sense, he eqaon 7 ves o: H D Fom he oh eqaon o 5, a solon non s avalable as dsssed n pevos sb-seon Consevave Feld Mae Moon Eqaon. Fo he smples solons n om, one has: D q σ D H ρ J whee: D J ρ [ ] ρ [ B B] To make he poblem smple, Colomb ae:, o B s sed o deny he seady ees om he dynam poess. Based on hs saey, when he en densy s noded, he mae ms be deomable. Fo he eneal ases, hee s an addonal moon eqaon []: I an be ewen as: Noe ha one denes [ ] A B 5 6 A as sala poenal o ele eld, B as he veo poenal o mane eld, he eqaon 6 s he Loenz ae. Ths eqaon an be esablshed om he ona-ovaan om o mae moon eqaon deved om he leas aon pnple o he non-ommave eld. Theeoe, Mawell eqaons and boh Loenz ae and Colomb ae [] ae deved n hs eseah. 9
The above eqaons om he Mawell eleomane eld eqaons. b. Coheen Lh I s hamon abo me wh eqeny ω and wave nmbe k, one wll has: A ω, B k 7 Pn hem no eqaon 7, one nds ha, o,, : ω k ωk ωk 8 Tha s: ω k ωk 9 The nns eae o wave nmbe-eqeny elaon s ha s ompleely deemned by he hae and mass ao. In physs, o bas pales, he hae-mass ndeed plays as bas paamee. Ths s lea om ollown dssson. I ω s a spaal onsan, hen one has: k ρ ωk ωk 5 ρ I has a solon: ρ k k ep ω 5 ρ I shows ha he wave nmbe deeases o zeo o lon me daon. I shows ha he mae eld s loalzed abo me. I an be sed o dene he le-daon o bas pales. I k s a me onsan, hen one has: q ω ρωk 5 I has a solon: ρ ω ω ep k 5 q I shows ha he mae eld s loalzed abo spae, so o sh a knd o mae he lm onaon an be dened. Fo he lh wave n vam, as he physal obsevaon has poven ha he wave nmbe-eqeny s lnea elaon, based on he above eqaons, hee ms be eqaon:, o q ρ 5 I means ha he lh s a speal mae n ha he addon o s Newonan mass and s Colomb ele hae s zeo. In a, hs onlson s e o he eleomane wave n vam. The eqaons 8-5 n a an be sed o eplan he heoe bases o oheen lh o he oheen wave poess o eleomane wave [].. Pe Mane Feld and Geo-Mane Feld A I o, ha s he A s a pe me non, one wll e: ρ B
B ρ 55 ρ B I denes mane eld. o, one an say ha he me aden o he spaal dsbon o me dsplaemen deemnes mane eld. When he spaal oe has voy, hen wll be he soe o mane vaaon. Tha s: H ρ 56 I shows ha he voy o Newonan spaal oe s popoonal wh he mane eld me devave and he oeen s he Newonan mass. Ths eqaon an be sed o eplan he Eah mane eld vaaon and he soe o he Geo-mane eld. The esene o mane eld o sas shold be nomal phenomena. Ths eqaon an be heked by he elaed obsevaon n spae physs, espeally he n mane eld.. d. Pe Ele Feld B I o,,, ha s he B s a pe spaal veo-non, one wll e: q A A q 57 q A q Ths eld oesponds o a onsevave eld n o-dmensonal oodnao sysem. A Who osde soe, he vaaon o osm oe wll pode an addonal ele poenal. Ths addonal ele poenal wll pode spaal oes. I one denes hs as weak aon oe, he eld desbed by eqaon may be dened as son aon. mmn p above dssson abo me dsplaemen eld, one nds ha avy eld, eleo-mane eld, and qanm eld ae me aden eld. They ae elaed wh Newon s mehans n nns sense. o, he ned eld heoy o mae moon an be epessed by he ne eomeal eld heoy ven by hs pape. 5. Geomeal Invaans and Consevave Felds n Classal Physs Fo a mae, s dened nally n vam eomey as havn eome nvaan: d d d d ds 58 Then, he mae moon n pely me dsplaemen sense wll have en eome nvaan as:
] [ ] [ ] [ d d d ds d d d d d d ] [ d 59 Hee, s he veloy o neal sysem. The physal eome nvaan s: d d d ds ds d d d d d d ] [ d 6 o, eome nvaan o physal moon an be dened as: d d d ds ds d Σ 6 whee: d d d d d d d d d d d 6 Fom physal onsdeaon, no mae wha nal eomey sysem s sed as nal o-movn oodnao sysem eomey, he sbevy o physal moon wll eqe he Σ d be eomeal nvaan. I an be dvded no hee smple ypal ases, as desbed bellow. a. Non-Qanm Mae Moon Newon Mass-pon Based on he pevos eseah, o non-qanm mae moon he me dsplaemen eld omponen an be eplaed by Newon s Aeleaon Law. I s eqvalen o sppose hee s no me dsplaemen. Hene, we sppose o non-qanm mae moon: d >> 6 Hene, he eomeal nvaan o non-qanm mae moon s: d d Σ 6 Tha s: d d d d Σ 65 Ths denes he onvenonal hee-dmensonal physal mease spae. I denes he spaal moon dsane as eomeal nvaan. Fo Newon s mae pon, leads o he kne eney onsevaon. By eqaon 6, o mass ρ dsbon, he spaal neal: Ω Ω Σ d d d d W ρ ρ 66 s eomeal nvaan. I leads o he kne eney onsevaon: dω w ρ 67
Ths s he adonal Newonan mehans. The spae s Caesan and he me s an ndependen paamee. b. ey Hh peed Moon o Newonan Mae Fo vey hh speed mae moon, he me aden o me dsplaemen an be noed, ha s o say: Then we wll have he eomeal nvaan as: >> 68 d Σ ds ds d d d d 69 whee: d d d d d d d d d d 7 I an be seen ha d s ndependen wh d, so s a sala dened n hee-dmensonal spae. Fo mae behaves as he me dsplaemen s dened by he ollown wave om: U ep[ R k ± ] 7 n o mae behaves as he me dsplaemen s dened by he ollown spaal-loalzed hamon pale om: U ep R l ± 7 n I we ake he denon o: d dτ 7 and eplan he d τ as he nvesal me wold me, we wll e he eomeal nvaan as: d Σ d d d d 7 Tha s: d Σ d d d dτ 75 In eneal elavy heoy and n ae eld heoy, hs eqaon om s sed o dene he eomey o spae-me onnm nde he meann ha he d Σ s he eomeal nvaan o wod-lne dsane. Howeve, based on hs pape s eseah, hs s only e when he me dsplaemen s nomal and n om 7 o 7. Fo moe omplaed mae moon, he ae eld heoy and eneal elavy wll be aled. mmn p above dssson, we an see ha adonal physal onsevaon laws and he eomeal nvaan noded by he ae eld heoy and eneal elavy ae nlded n he ne eomeal eld heoy as smple speal ases.. Consevave Felds Based on he pevos eseah, o qanm mae moon he me dsplaemen eld
omponen play he man oles. I s eqvalen o sppose hee s a snan me dsplaemen. Hene, we sppose o qanm mae moon: << Hene, he eomeal nvaan o non-qanm mae moon s: d 68 Σ ds ds d 69 d o, we an dene me dsplaemen s a sala non n o-dmensonal me-spae. As a speal ase, hs eomeal nvaan an be eplaned as mae-eney onsevaon. In he pape Ineal ysem and peal Relavy [], he onsevave eld avaon eld and sa ele eld has a ypal solon n om: M 7 whee, he zeo pon s sally dened by he ene pon,, o he mae; M paamee may be elaed o Newonan mass o he mae o he sa hae o he mae, o he smmaon, whh depends on he op. Hee, s smply eeeed as eneal hae. In hs ase, he me dsplaemen s he only one non-zeo dsplaemen omponen. Aodn o he moon ansomaon n s ase een eqaon, one has: 7 Fo smply,, ae no aken no onsdeaon. Ths op wll be lly eaed n he pa hee pape o hs eseah, whee qanm mehans s he man heme. Fo an neal sysem dened on he onsevave mae and he onsan eneal hae,, pn he eneal solon 7 no he eqaon 7, one has: M M M 7 M M M In a omn he hhe ode nnesmals, he appomaon s:, M 7
Fo avy eld, he eqaon 7 s appomaon o he ypal solon hwazshld me. The eqaon 7 s he appomaon o Ke me. To make hs pon lea, le s onsde he eqemen o neal sysem. Fo hs ase, he physal me sale s dened by:, o make he spae sale be he absole same no dependen on, he spae sale ms be adsed o make: s he absole same. o, he en ae ha mees he denon o neal sysem and he onsevave eld eqaons an be dened as: M M, Howeve, o he osm aes old evens, he spaal vae may be anno be noed 7 when he ollown he s vey b, ha s: M ϑ 75 On hs sense, he oldes avy eld has he laes vae o spae. Ths pon s vey nns o asonomy maes. Theeoe, s easy o ndesand ha he eneal elavy heoy s well aeped by asonomy physs. Unonaely, he ees ased by he osm ae dependen vae o spae ae no aken no onsdeaon o he obsevaon osms o asonomy physs. Ths may be he man ase o nodn he B-ban heoy and s elaed alenaves. Fo he mo-wold mae, when he ele hae s onened, he old mae sll has a mh vae spae. Ths makes he qanm mehans beome he mos omplaed physal wold, whee he eneal elavy sll play an mpoan ole. Ths wll be he man op o ne pape on ne eomeal eld heoy. 6. Conlson When he mae has no mao spaal moon, sh as avy eld o eleon eld, he me dsplaemen eld s noded o desbe sh a knd o mae moon. Fo ommave ase, s onded ha avy eld, eleo-mane eld, and qanm eld ae me aden eld. The absole eleal hae qany may be epessed as q /. When hee s no eleal hae qany, he avy mass and he neal mass n Newon s mehans ae he same. The mae has hee ypal esn oms: aveln wave, loalzed hamon vban, and eponenal epandn o deayn. The ommable mae moon denes onsevave eld. The ommave mae moon denes a qanm eld hae ρ / dened as mae omplee qaly. Fo non-ommave mae moon, he Mawell eleomane eld eqaons ae esablshed, based on he ovaan om o mae moon eqaon deved om he leas aon pnple. When he ona-ovaan s eqed o he non-ommave eld, boh Loenz 5
ae and Colomb ae ae deved n hs eseah. The pape shows ha he lh s a speal mae n ha he addon o s Newonan mass and s Colomb ele hae s zeo. In a, hs onlson s e o he eleomane wave n vam Fo he onsevave eld, he eseah shows ha one he mass densy and he Colom hae dendy ae ven, he mao spaeme eae s ompleely deemned.boh o hem ae nsn eaes o mao mae n osm bakond. Howeve, o he osm aes old evens, he spaal vae may be anno be noed when he ollown he s vey b, ha s: M ϑ 75 On hs sense, he oldes avy eld has he laes vae o spae. Ths pon s vey nns o asonomy maes. Reeene [] Xao Janha. Ineal ysem and peal Relavy Fne Geomeal Feld Theoy o Mae Moon Pa One. axv: physs/5, 5, -9 [] Chen Zhda., Geome Theoy o Fne Deomaon Mehans o Connm. Aa Mehana na, 979, No., 7-7 In Chnese. [] Chen Zhda., Raonal Mehans. Chonqn: Chonqn Pblaon, In Chnese. [] Ensen, A., Relavs Theoy o Non-ymme Feld, In Ensen s Wok Colleon ol., Commeal Pb., 977, p56-565 n Chnese. Onal: Ensen, A., Meann o Relavy, 5 h edon, 95, p-66. [5] Ensen, A., Fondaon o Geneal Relavy, In Ensen s Wok Colleon ol., Commeal Pb., 977, p9 n Chnese. Onal: Ensen, A. De Gndlae de Allemenen Relaväsheoe, Annalen de Physk, -9, 96, p769 8. [6] Dh, W., M. Ree. Classal and Qanm Dynams eond ed., New Yok: pne-ela, 996 [7] Lopes J. L., Gae Feld Theoes, Peaman Pess. 98 [8] Wan Yn, Physs o Blak-hole, Chansha: Nomal Unvesy o Hnan, In Chnese. [9] Dbovn, B. A., A. T. Femenko,. P. Novkov, Moden Geomey Mehods and Applaon, Pa I: The Geomey o aes, Tansomaon Gops and Felds. New Yok: pne-ela, 98 [] Pao Yh-Hsn. Eleomane Foe n Deomable Connm. Ben: en Pblaon, 996 In Chnese Tanslaon. p [] R. Lodon. Opal Phenomena. Ben: Hhe Edaon Pblaon, 98 In Chnese Tanslaon. p [] He Xana, Obsevaon Cosms, Ben: en Pblaon, In Chnese. 6