Origin of the inertial mass (II): Vector gravitational theory

Transcription

1 Orn of he neral mass (II): Veor ravaonal heory Weneslao Seura González e-mal: Independen Researher Absra. We dedue he nduve fores of a veor ravaonal heory and we wll sudy wheher hese fores an be denfed wh he fores of nera ha a on a body when s aeleraed. 1. Inroduon In a prevous wor we nvesae he nduon phenomena derved from a salar ravaonal heory 1. We fnd ha here are fores ha have her orn n he movemen of he feld soure, bu hey do no sasfaorly explan he orn of nera Now we propose a veor ravaonal heory and we wll also nvesae s nduve effes, wh he dea of explann he phenomenon of nera. An nvesaon of hs ype was made n a famous wor by Sama.+ When sudyn he effe of he Unverse as a whole, we have o hoose a osm model. As we are neresed n he qualave aspes, we onsder he smples osm model: sa, wh homoeneous densy, lare enouh and of fne ae. In laer wors we wll examne he problem wh more reals osmoloal models. We advse he prevous readn of he frs par of hs researh: «Orn of he neral mass (I): Salar ravonal heory» 1. Veor ravaonal heory To oban a veor ravaonal heory we have o adap he eleromane heory o he ravy 3. The avaonal poenal s a eraveor ha we defne by, A s he salar poenal and A he veor poenal. The soure s he urren eradensy s he proper densy of maer and d s he proper me of he soure parle. The feld equaon s j u j o, j u s he eraveloy defned by u dx d 1 4 G j 3 The proper me We assume ha movemen s relave. In oher words, s equvalen o say ha a body moves wh respe o he Unverse, ha o say ha s he Unverse ha moves wh respe o he body. Boh suaons are equvalen nemaally and dynamally. In our smplfed osm model all he onsuen bodes of he Unverse are a res amon hemselves, wh he exepon of he C body whose movemen we sudy. 1. (1) ()

2 Seura González, Weneslao When we assume ha he Unverse as a whole s n moon, he proper me of s onsuens ondes wh he oordnae me of he referene sysem wh respe o whh he Unverse s a res, whh s naurally an neral referene sysem, where all los ha are a res an be synhronzed. Therefore, he eraveloy of any fxed body o he whole of he Unverse n any referene sysem s herefore u dx dx d dx u u u j j u d Then he feld equaon s dvded no a salar feld and a veor feld d, ; ; ;. 1 1 A 4 G 4 G ; A j. 4 Rearded poenals Equaons (3) are solved wh he rearded poenals ehnque G j r, G dv ; A r, dv r r (4) r s he poson veor of he feld pon, r s he poson of he feld pon wh respe o he soure a he momen of eneran he fore (or poson rearded) and he braes are values a he momen of emsson or rearded values, ha s n he nsan r.the veor s he poson of a soure pon, herefore r r σ. From (4) we noe ha veor poenal s of seond order wh respe o. If he soure moves wh a veloy u hen he poenals of Lénard-Weher have o be appled dm dm G u d G G ; da dm. u r r s s For he alulaon of (5) we mus use relaonshps 4 1 r u r ru r ra u r s s r s s r s s u r a u ru u ra uu. s s s s r s s u s he rearded veloy of he soure and a he aeleraon, herefore σ r u d d. d d The ravoeler feld E and he ravomane feld B are derved from he poenals and A n he same way as n eleromanesm A E ; B A, he fore an on a parle of ravaonal mass m ha has a veloy w s F m E m w B. 5 Induon of fores on a body ha has a relnear aeleraon Suppose a body C ha moves wh aeleraed and relnear movemen wh respe o he whole of he Unverse. The suaon s equvalen o sayn ha he Unverse moves wh respe o he body C. u and a are he veloy and aeleraon of he Unverse wh respe o he body C. Therefore he body C has veloy u and aeleraon a n relaon o he Unverse. To nerae (5) we assume he Unverse dvded no onenr spheral shells of hness hp://vxra.or/abs/ (3) (5) (6) (7) (8)

3 Orn of he neral mass (II): Veor ravaonal heory 3 d. dm s an nfnesmal poron of mass of one of hose shells, hen he fore exered on body C s dedued from (5) and (8) da 1 G u m d G dm m dm m. s s Comparn (4) and (5) and assumn non-relavs veloes herefore ru dm 1 dv dv, r da 1 G u m d G m dv m. dv s s From (6) we fnd ha he only nduve erms (whh are hose ha depend on he movemen of he soure) are 1 r ra σ σ a u a u ra a u σ a ; 3 3 s s s s s s s s we have pu r s and we nele he erms ha depend on 1 beause hey are nelble ompared o hose ha depend on 1. For smply we assume ha he aeleraon a of he Unverse s parallel o he x axs a a and ha n polar oordnaes he veor s σ sn os sn sn j os (9) usn he resuls found n he referene 1 we arrved afer he neraon o A G G 4 m m a a a m G m (1) 3 3 he lm of spaal neraon s he furhes pon from he Unverse ha has a ausal relaonshp wh he body C, ha s, where s he ae of he Unverse. The fore (1) as on he body C when has he relnear aeleraon a, and s aused by he aon of he whole of he Unverse. We denfy hs fore wh he fore of nera, herefore 4 4 m a G m a m G m, (11) 3 3 m s he neral mass. As n he presen momen he neral mass s assumed equal o he ravaonal mass, hen by (11) G (1) s he urren ae of he Unverse and s he urren ravaonal poenal n he poson ouped by he body C. Naurally, hese resuls are vald n he framewor of he osm model onsdered. (11) means ha he neral mass s he resul of he aon of he whole of he Unverse on he body, s no an nnae manude of he maer, bu s aqured by he osm aon. The numeral value (1) s approxmaely orre. Therefore we onlude, whn he lmaons of our osm model, ha he Mah's prnple s fulflled, whh saes ha he nera of a body s produed by he nduve fores of he Unverse. 6 Induon fore on a body n unform moon Suppose a body C ha has a unform and relnear movemen wh respe o he Unverse. u s he veloy of he Unverse wh respe o C. Nex we alulae he nduon fore ha s exered on C. The only erms of (6) ha we have o use n hs problem s he seond and hrd of he frs equaon, he oher erms no null an be neleed for non-relavs veloes, hen 1 u r ru 3 s s s r and he nduon fore produed by an elemen of he spheral shell s hp://vxra.or/abs/189.53

4 4 Seura González, Weneslao da u r ru m d Gm sn ddd. 3 s s r Ineran for he whole of he Unverse and nelen erms, he nduon fore an on he body C s 8 F G m u, (13) 3 by he prnple of dynam equlbrum he sum of all he fores an on he body s zero. We assume ha on he body as n addon o he fore (13), he fore of nera, whh for a body n relnear moon s he equaon (1), hen when we apply he prnple of dynam equlbrum s found 8 4 u F F Gm u G m a a (14) 3 3 s he ae of he Unverse and a s he aeleraon of he body C. The equaon (15) ells us ha when a body s n unform moon wh respe o he Unverse, an nduon fore appears ha produes an aeleraon ven by (14), of manude so exraordnarly small ha we an onsder null. Therefore here s no fore of nera on a body n unform moon and herefore remans wh equal veloy, a saemen ha orresponds o he prnple of nera. 7 Parpaon of he Unverse n he formaon of he neral mass Aordn o Mah's prnple, he neral mass of a body s eneraed by he aon of he whole Unverse. On a body C a fores ha are produed n all pars of he observable Unverse. Some of hese fores ornae n very dsan objes, ha s, hey were produed a lon me ao; whle he fores exered by nearby objes were produed a shor me ao. Now we sudy he parpaon n he eneraon of he neral mass of he dfferen epohs of he Unverse. We onsder he Unverse formed by N spheral shells n whose ener s he body C. To he furhes shell we ve he numeraon n and n are he objes ha produe he fore a he bennn of he Unverse, whose effes reah he body C n he presen momen, when he ae of he Unverse s. The hness of he spheral shells s where N. The neral mass produed by shell n s by (11) 8 m Gm (15) 3 s he me aes he ravaonal neraon o ravel from he spheral shell n o he body C, herefore of (11) and (15) we e n m 1 (16) m N n s he me rearded. We mus remember ha (16) s applable o he osm model ha we are onsdern whh has a onsan and unform densy of maer n he Unverse. Bu ndependenly of he osm model, (16) shows us ha eah epoh of he Unverse onrbues dfferenly o he formaon of he neral mass of a body. (16) shows ha he frs momens of he Unverse (when ) have a reaer onrbuon o he neral mass; whle he presen Unverse ( ) he onrbuon s mnmal. Ths resul an be more pronouned n a reals osm model. 8 Mah's prnple and he B Ban Now we wll use a more reals osm model wh he exlusve dea of sudyn he nfluene ha he B Ban has on he formaon of he neral mass, and onsdern manly he qualave aspes. Now we suppose, for example, a Unverse where he densy depends on he osm ae hp://vxra.or/abs/189.53

5 Orn of he neral mass (II): Veor ravaonal heory 5 aordn o he law (17) s he densy a he presen momen and s he ae of he Unverse a he me of eneran he fore or rearded me. By (17) he densy of he Unverse would be very hh when s ae s small, a suaon smlar o wha happens n he B Ban, herefore (17) wll nform us abou how he neral mass s eneraed n a Unverse ha has a hh densy a s bennn. By (16) he onrbuon o he neral mass of a spheral shell loaed a a dsane and hness d s 8 8 dm G m d G m d 3 3 when we nerae for he whole Unverse we fnd a snulary n. To avod hs snulary we alulae he aon of he Unverse from a me (nsead of ) m 8 G ln1, (18) m 3 wh he urren values of and s found 8,9. 3 G we verfed ha he neral mass observed n he urren epoh would be eneraed by he fores produed by he Unverse from,856. Bu an no aoun anen osm aons, he relaonshp (18) nreases, whh means ha he neral mass beomes muh reaer han he ravaonal mass. For example, f we ae he lm a,999, whh orresponds o an ae of he Unverse of abou foureen mllon years, he rao beween he neral mass and he ravaonal mass s approxmaely 5,4. The onluson s ha f he densy n he pas was reaer han a presen (as mus happen n a osm model wh B Ban), he parpaon n he neral mass of he prmve Unverse nreases, unl ben deermnan. Tha s o say, ha he neral mass would be eneraed manly by he frs saes of he Unverse, unl ben very superor o he mass observed. We onlude ha f he Unverse had a B Ban, he Mah s prnple would no be fulflled and ve versa. 9 Fore ndued by osm expanson Nex, we alulae he nduon fore produed by a movemen of osm expanson. We assume ha he observer s a he ener of he Unverse, herefore r. And ha he expanson veloy s radal σ u u u sn os u sn snj u os where u s any funon of. We follow he same proedure as n he prevous seons. So when alulan he anular nerals ha are derved from he wo equaons (6) we fnd ha he resul s zero. Tha s, a osm expanson does no produe nduon fore, whh s loal for reasons of symmery. 1 Conlusons We have used a veor ravaonal heory, from whh we dedue nduon fores. The resuls show ha s possble o denfy he nduon fore produed by he Unverse wh he fore of nera an on an aeleraed body. We have verfed ha here are no fores of nduon on a body ha has a unform movemen, as s requred by he prnple of nera. We verfy ha he osm expanson does no produe nduon fores. We have found nompably beween Mah's prnple and a osm model wh B Ban. hp://vxra.or/abs/189.53

6 6 Seura González, Weneslao In hs model he neral mass of a body s manly produed by he frs momens of he Unverse and he eneraed mass snfanly exeeds he observed neral mass. The onluson s ha alhouh he veor ravaonal heory s an approxmaon o he orre heory, and he osm model ha we have used s very smple, we fnd ha he nduon fores produed by he whole Unverse are omparable n sense and manude o he fores of nera ha are observed n aeleraed bodes, whh s a sron suppor for Mah's prnple. Bbloraphy (1) SEGURA GONZÁLEZ, Weneslao: «Orn of he neral mass (I): salar ravaonal heory», vxra.or, 18, hp://vxra.or/abs/ () SCIAMA, D. W.: «On he orn of nera», Monhly Noes of he Royal Asronomal Soey, Vol. 113, 1953, pp (3) SEGURA GONZÁLEZ, Weneslao: Teoría relavsa de ampo, ewt Edones, 14, pp (4) SEGURA GONZÁLEZ, Weneslao: «Mah s prnple: he osm orn fo he neral mass», 17, vxra: (5) PANOFSKY W.; PHILLIPS, M.: Classal Elery and Manesm, Addson-Wesley, 197. pp (6) SEGURA GONZÁLEZ, Weneslao: Gravoeleromanesmo y prnpo de Mah, ewt Edones, 13, pp hp://vxra.or/abs/189.53