Origin of the inertial mass (I): scalar gravitational theory

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1 Orn of the nertal ass (I): salar ravtatonal theory Weneslao Seura González e-al: Independent Researher Abstrat. We wll dedue the nduton fores obtaned fro the feld equaton of Nordstro's salar ravtatonal theory and nvestate whether they an explan the orn of the fores of nerta atn on a body when t s aelerated. 1. Introduton The General Theory of Relatvty s a relatvst feld theory, whose potental s the seondorder etr tensor. The theory s expressed wth a syste of nonlnear dfferental equatons of seond deree. The soure of ths feld s the seond-order enery-oentu tensor. But we an nvestate spler ravtatonal theores, suh as a salar theory (whose potental s the ealar ) or a vetor theory (whose potental s the tetravetor ). In the frst theory the feld soure s a salar (the densty of ass ) and n the seond theory t wll be a vetor (the j ). Both theores have to be developed wthn the fraewor of the speal urrent densty theory of relatvty, that s, the spae-te s pseudo-euldan. These theores are lnear, whh eans that t s neessary to add the equatons of oton. Ths s not neessary n a non-lnear theory suh as General Relatvty. In ths paper, we wll study the ndutve effet of salar ravty and nvestate whether the nduton produed by the whole unverse an explan the orn of nerta. In the year 195 Denns Saa wrote a faous paper enttled "On the orn of nerta" 1,, where he developed a splfed vetor ravtatonal theory and affred that a salar theory of ravty would not produed ndutve effets. As we wll see later ths stateent s wron. Mah's prnple affrs that the nerta (or the nertal ass) of a body s ornated by the whole of the Unverse. We understand that the fore of nerta s a where s the nertal ass and a the aeleraton of the body C, that s, a fore that s opposte to the aelerated oveent of C. Then the Mah's prnple affrs that ths fore of nerta s produed by the ndutve fores that have ther orn n the relatve oveent of the Unverse wth respet to the body C. We try to prove whether ths stateent s true. Newton's ravtatonal theory as a feld theory. Nordströ theory Althouh n ts ornal for Newton's theory of ravtaton s not a feld theory, we an forulate t as f t were. In effet, f s the ravtatonal potental, the feld equaton would be the Posson equaton wth the equaton of oveent 4 G a and are the nertal and ravtatonal asses of the body on whh ravty ats and a s ts aeleraton. But the equaton (1) s not nvarant under Lorentz transforatons, whh eans that the feld equaton depends on the referene syste, what s not adssble. The feld derved fro (1) s transtted at nfnte veloty, as orresponds to the aton at a dstane of the theory of Newton. (1) () 1

2 Seura González, Weneslao However, f we odfy (1) as 1 4 G () t then we have a feld that s transtted at the veloty of lht. Sne and are salars, the feld equaton () s a relatvst nvarant. We therefore have a salar ravtatonal theory opatble wth the speal theory of relatvty. () s the feld equaton of Nordströ's salar theory The equaton () ust also be eneralzed. As the lnear oentu s defned by w p w 1 w then the fore atn on the partle of ass that has a veloty w s F d w dt for nonrelatvst velotes ondes wth (). Retarded potental. Potental of Lénard-Wehert The soluton of equaton () s obtaned by the theory of retarded potentals t r r, t G dv G dv r (4) r the vetor r s the poston of the pont of the feld, r s the retarded poston of the soure, t s the oent n whh the ravtatonal snal arrves at the pont of the feld, dv s the eleent of volue ouped by the soure and the braets are values at the oent of esson or retarded values, that s n the nstant t t r If the soure of the feld has a veloty u and an aeleraton a, equaton (4) ust be adapted. The ravtatonal feld d produed n the poston r and at te t by a ass eleent d that has veloty u, aeleraton a and s n the poston r s ven by the potental of Lénard-Wehert d d d G G. u r r s To apply the equaton (5) to the salar ravtatonal theory we need to now that 4 s defned by 1 r u r ru r ra u r s s r s s r s s j x y z x, y, z are the Cartesan oordnates of the feld. In equaton (6) appears the veloty and aeleraton of the soure, then the potental wll also depend on ths veloty and aeleraton, therefore n the salar ravtatonal theory there are phenoena of nduton, that s fores produed by the oveent of the soure. 4 Relatve oveent In the study that we wll do next we aept that the oveent s relatve. Only the oveent of one body wth respet to another body has eann, and all the antudes that are used (poston, veloty and aeleraton) are relatve to other bodes. Therefore we deny the exstene of a Newtonan absolute spae. It s neat and dynaally equvalent to affr that a body oves wth respet to the whole of the Unverse or to say that t s the Unverse that oves wth respet to the body. Now, the relatve oton of the Unverse produes ravtatonal ndutve effets as verfed by (5) and (6), fores that at on any body that oves wth respet to the Unverse. Aordn to Mah's prnple, these fores are the fores of nerta. We wll study the ndutve fores that are enerated n the salar ravtatonal theory for (5) (6)

3 Orn of the nertal ass (I): salar ravtatonal theory several dfferent stuatons. 5 Induton of fores on a body that has a retlnear aeleraton Let's onsder a splfed odel of the Unverse, whh wll ve us an aeptable vew of the proble. We wll assue that the Unverse s stat, of unfor densty, of suffently lare sze and lted ae. All the bodes of ths Unverse are at rest. A body C of ravtatonal ass has an aelerated retlnear oveent wth respet to the whole of the Unverse. As we onsder that the oveent s relatve, the stuaton s equvalent to suppose that the body C s at rest and t s the Unverse that oves n the opposte sense of the body C. a and u are the aeleraton and veloty of the Unverse wth respet to the body C. Whh eans that C oves wth aeleraton a and veloty u wth respet to the Unverse. K s a oordnate syste whose orn O s lose to body C and s at rest n relaton to the Unverse. The poston vetors are those of drawn 1, therefore r r σ. Feld pont (7) r r O Drawn 1. Soure pont We want to obtan lassal effets, therefore we wll suppose that the veloty u s very sall wth respet to, that s, we nelet the relatvst effets. Then we onsder the os te t, the sae for the whole Unverse. To ae the nteraton of all the ndutve fores produed by the Unverse we wll onsder that t s fored by onentr spheral shells of "nfntesal" thness d. d s an eleent of ass belonn to one of those shells, then the fore that ths ass exerts on the body C s by (5) 1 d G d. s Sne the veloty u s sall wth respet to, then ru s r r. Fro (6) we fnd that the only of ts ters that we have to onsder s the fourth, that has the dependene 1 ; all the others ters have the dependene 1 and sne has a os denson these ters an be dsarded. The frst ter of (6) produes the Newtonan ravtatonal fore that we now vanshes when t s nterated for the whole Unverse. Coparn (4) wth (5) ru d 1 dv dv, r usn spheral oordnates we have of (8) 1 σ σ a d G dv G sn d d d. s The nduton fore exerted by the spheral shell of thness d on the body C s G σ σa d sn ddd (9) s the densty of the Unverse that s onstant and hooeneous. Frst we wll do ths nteraton and then nterate for all the spheral shells of the Unverse. (8)

4 4 Seura González, Weneslao If, for sae of splty we assue that the aeleraton a of the Unverse s parallel to the x axs a a and that n polar oordnates the vetor s σ sn os sn sn j os (10) t s easy to fnd fro (9) 4 G. a d (11) Fnally we do the nteraton for the whole Unverse, tan nto aount that the furthest spheral shell that affets the body C s at a dstane t, ben t the ae of the Unverse, sne the fores on fro ore dstant bodes have not had te to reah the body C. The nteral of (10) results G. a t (1) (1) eans that f the body C of nertal ass has an aeleraton a wth respet to the Unverse, on t ats the nduton fore (1), whh an be understood as the fore of nerta, naely a G at Gt (1) whh s absurd, not only beause the nertal ass would be neatve, but beause the fore of nerta would have the sae sense as the aeleraton of the body C, whh does not orrespond to the observed. In the follown setons we wll nvestate whether wthn the fraewor of the salar ravtatonal theory nertal fores are enerated that we an dentfy wth the entrfual and Corols fores. 6 Centrfual fore nduton We wll nvestate f the relatve rotaton of the Unverse ndues entrfual fore on a body C. For ths we use (5) and (6) and follow the sae tehnque as n the prevous seton. We assue that body C s on the z-axs r r we assue that the rotaton of the Unverse s around the y axs ω j, whh eans that the body C rotates wth respet to the Unverse wth the anular veloty ω. The vetor of poston of a pont of the Unverse s ven by (10). The veloty of a pont n the Unverse wth respet to the oordnate syste for whh the Unverse s rotatn s u ω σ os sn os ; a ω ω σ sn os os As σ u 0 then t t 1 u r r 1 r ω σ t t. ben t the proper te at the orn of the oordnate syste, or the teporal oordnate n the nertal syste wth respet to whh the Unverse s at rest. r and are proper dstanes, whh onde wth the oordnates dstanes sne radal dstanes are not affeted by relatvst effets. s perpendular to u then by (7) ru r sn os r sn os s r r 1. The only ters n (6) that are not aneled when the nteraton s done are the fourth and the ffth. Then the fore ndued by an nfntesal spheral shell s at a dstane 1 r r a u r d G dv G sn d dd. s s s

5 Orn of the nertal ass (I): salar ravtatonal theory 5 Interatn for the whole Unverse then t follows 4 G rt G rt G t, (14) we fnd aan a fore of nerta ontrary to the observed, whh s always opposte to the aeleraton of the body. However, the nueral value of proportonalty between the nertal and the ravtatonal ass s rouhly n aordane wth the os easureents. If the Unverse s supposed to be rotatn t ust be adtted that ts onsttuents, exept the losest bodes, have a veloty reater than that of lht. But these superlunal velotes are not a proble beause the whole Unverse oves n unson and therefore the prnple of ausalty reans vald. There are no snulartes n expressons le 1 u sne the proper te of all the onsttuents of the Unverse s the proper te of the orn of the referene syste or the oordnate te of the referene syste wth respet to whh the whole Unverse s at rest, whh s obvously an nertal referene syste. 7 Corols fore and fore produed by anular aeleraton The Corols fore depends on the veloty w of the partle that oves wth respet to the Unverse. As n (6) ths veloty does not appear, t s possble for the Corols fore to be dedued fro the nduton produed by the potental Suppose that the body C s loated n the poston r near the orn of oordnates and that t has a rotaton aeleraton ω. Ths stuaton s equvalent to the fat that body C s at rest and the Unverse has an anular aeleraton ω. Let's nvestate the ndutve fore of ths oveent. The only ter that an enerate the fore of nerta produed by the anular aeleraton s the fourth ter of (6), therefore It s easy to he and the nteral (15) ves G r ra sn. d dd r r a σ r a r sn os G t r (16) the aeleraton of C wth respet to the Unverse s ω r r. Now we dentfy ths fore (16) wth the nertal, resultn Gt ths s an absurd result beause t would ean that the fore of nerta of the body C would have the sae sense as ts aeleraton. Therefore the salar ravtatonal theory s unable to explan the fore of nerta on a body that has an anular aeleraton. 8 Conlusons We use a salar ravtatonal theory that s relatvst nvarant. The feld equaton s solved by the tehnque of retarded potental. We also assue that oveent s relatve, that s, the bodes ove n relaton to other bodes not n relaton to a hypothetal spae. We onsder a very sple os odel, where the densty of atter s onstant and hooeneous. We are not nterested n the exat nueral results, we are ore nterested n the qualtatve aspets of the proble. In the prevous nvestaton we have verfed whether the fores of ravtatonal nduton produed by the relatve oveent of the whole of the Unverse enerate the nertal fores that are observed, or n other words f the phenoenon of nerta has an orn n the osos. (15)

6 6 Seura González, Weneslao The results obtaned are the follown: 1) Salar ravtatonal theory produes ndutve effets, that s, t enerates fores dependent on the aeleraton of the ravtatonal feld soure, ontrary to what Saa had predted. ) The nduton fores that have been dedued are of opposte sense to the fores of nerta, therefore ontrary to observaton. ) The nueral values of the nduton fores derved fro the salar ravtatonal theory are approxately orret 0 1. G t where t 0 s the urrent ae of the Unverse 4) The ravtatonal ass of a body has an nalterable value, but the nertal ass s varable and deterned by the ravtatonal ass and by the os propertes of the Unverse. Ths varablty wth te of the nertal ass has easurable effets. 5) The nertal ass s proportonal to the ravtatonal ass, as experentally observed. 6) There s no fore of nerta when the body has an unfor oveent, as the law of nerta affrs. 7) It s found that the nduton fore that we dentfy wth the fore of nerta depends lnearly on the aeleraton, at least n the lassal approxaton, as ndated by Newton's seond law. In the seond part of ths study we wll alulate the ndutve fores derved fro a vetor theory of ravtaton, we wll see that the results wll be uh better than those found for the salar theory. 9 Bbloraphy (1) SCIAMA, D. W.: «On the orn of nerta», Monthly Notes of the Royal Astronoal Soety, Vol. 11, 195, pp.4-4. () GRANEAU, P.; GRANEAU, N.: In the rp of the dstant Unverse, World Sentf, 006, pp () PANOFSKY, W.; PHILLIPS, M.: Classal Eletrty and Manets, Addson-Wesley, 197, pp and pp (4) SEGURA GONZÁLEZ, Weneslao: «Mah s prnple: the os orn of the nertal ass», 017, vxra: