On the Essence of Electric Charge Part 1

Transcription

1 On the Essence of Electc Chage Pat Shlomo Baa To cte ths veson: Shlomo Baa. On the Essence of Electc Chage Pat : Chage as Defomed Space. 06. <hal > HAL Id: hal Sbmtted on 3 Nov 06 HAL s a mlt-dscplnay open access achve fo the depost and dssemnaton of scentfc eseach docments, whethe they ae pblshed o not. The docments may come fom teachng and eseach nstttons n Fance o aboad, o fom pblc o pvate eseach centes. L achve ovete pldscplnae HAL, est destnée a dépôt et à la dffson de docments scentfqes de nvea echeche, pblés o non, émanant des établssements d ensegnement et de echeche fanças o étanges, des laboatoes pblcs o pvés.

2 On the Essence of Electc Chage Pat : Chage as Defomed Space Shlomo Baa Taga Innovatons 8 Bet Hllel St. Tel Avv Isael Coespondng atho: shlomo@tagapo.com Abstact The essence of electc chage has been a mystey. So fa, no theoy has been able to deve the attbtes of electc chage, whch ae: bvalency, stablty, qantzaton, eqalty of the absolte vales of the bvalent chages, the electc feld t ceates and the ad of the bvalent chages. O model of the electc chage and ts feld (Pat ) enables s (Pat ), fo the fst tme, to deve smple eqatons fo the ad and masses of the electon/poston mon/antmon and that of qas/ant-qas. These eqatons contan only the constants G, c, ħ and α (the fne stcte constant). The calclated eslts based on these eqatons comply accately wth the expemental eslts. In ths Pat, whch seves as a bass fo Pat, we defne electc chage densty, based on space densty. Ths defnton alone, wthot any phenomenology, yelds the theoy of Electostatcs. Togethe wth the Loentz Tansfomaton t yelds the ente Maxwell Electomagnetc theoy. Keywods: Electc chage, Space lattce, Electomagnetsm

3 . Intodcton In o model a postve elementay electc chage s a contacted zone of space, wheeas a negatve elementay electc chage s a dlated zone of space. Relatng to space as a lattce (cellla stcte), we defne Space Densty as the nmbe of space cells pe nt volme (denoted 0 fo space wth no defomatons). Based on ths we defne (postlate, nvent) Electc Chage Densty as: q = /4π (ρ ρ 0 )/ρ [q] =. Ths chage densty s postve f > 0 and negatve f < 0. Electc chage, n a gven zone of space τ, s then: Q qdτ [Q] = L 3. τ O defnton of electc chage densty alone yelds electostatcs, wthot any phenomenology, and togethe wth the Loentz Tansfomaton - the ente Maxwell theoy. Ths eslt encoages s to fthe pse o dea of the essence of electc chage and, as Pat shows, t yelds the mpotant eslts pesented below. Note that o defnton of chage densty s axomatc. Ths appoach s n the spt of Ensten [] that:. the axomatc bass of theoetcal physcs cannot be extacted fom expeence bt mst be feely nvented We consde chage to be a defomed zone of space, and snce the geomety of both defomed spaces and bent manfolds s Remannan, we can attbte cvate to an electc chage. Ths new dea, of chage as cved space, enables s to se the theoy of Geneal Relatvty (GR) n o devatons. These devatons, n Pat of ths pape, yeld the attbtes of matte. Some of whch ae pesented below: The poton chage ads, p s: p (calclated) = cm. well wthn the expemental eo ange [].

4 p (meased) = (69) 0-3 cm. Based on p, we calclate the electon ads e : e = cm. Based on ths e we deve and calclate the mass M of the electon: M(calclated) = g. A devaton of 0.06% fom the meased CODATA 04 vale: M(meased) = () 0-7 g. Based on ths M and o model of qas, we deve and calclate the masses M d and M ũ of the d and ~ qas: M d =4.5 MeV M ũ =.5 MeV ecent expemental vale [3]: M d =4.8 +/ 0.5 MeV ecent expemental vale [3]: M ũ =.3 +/ 0.8 MeV These eslts spea fo themselves, and stfy o axomatc appoach.. On O Electomagnetsm (EM) In ths pape we deve only Electostatcs, whch s needed fo Pat. The ente theoy of electomagnetsm and the extenson of the Geneal Relatvty feld eqaton to ncopoate the contbton of chage to cvate wll appea as sepaate papes. O EM tlzes the dmensons of length L (cm) and tme T (sec) only. Ths, each physcal qantty of o EM has a dmensonalty, whch s L x T y. Usng Geneal Relatvty we can establsh qanttatve eqvalence between o system and the conventonal system of nts. By elatng the feld enegy densty to chage densty o EM becomes non-lnea, whch s ts specfcty. Note that QED s also a non-lnea theoy. 3

5 The sses of stcte, stablty and qantzaton of an elementay chage ae dscssed n Pat. Note that o EM s applcable fo both a sngle elementay chage and an ensemble of elementay chages. Note also that EM waves, n o theoy, ae smply space tansvesal vbatons.. Space as a Lattce O defnton of Electc Chage Densty s based on the concept of space densty. Space densty s elated to the cellla stcte of space; elatng t to a contnm s not pohbtve bt poblematc. Attbtng a cellla stcte (a lattce) to space explans ts Hbble expanson, ts elastcty (see.3) and ntodces a ct-off n the wavelength of the vacmstate spectm of vbatons. Wthot ths ct-off, nfnte enegy denstes ase. The need fo a ct-off s addessed by Sahaov [4] and Msne et al [5]. The Beensten Bond [6] sets a lmt to the nfomaton avalable abot the othe sde of the hozon of a blac hole. Smoln [7] ages that: Thee s no way to econcle ths wth the vew that space s contnos fo that mples that each fnte volme can contan an nfnte amont of nfomaton. A evew, elevant to o dscsson, appeas n a pape by Amelno-Camela [8]..3 The Elastc Space We elate to space not as a passve statc aena fo felds and patcles bt as an actve elastc entty. Physcsts have dffeent, sometmes conflctng, deas abot the physcal meanng of the mathematcal obects n the models. The mathematcal obects of Geneal Relatvty, as an example, ae n-dmensonal manfolds n hype-spaces wth moe dmensons than n. These ae not necessaly the physcal obects that Geneal Relatvty acconts fo and n-dmensonal manfolds can be eqvalent to n-dmensonal elastc spaces. Ths eqvalence allows s to se Geneal Relatvty, and also elate to o own space as an elastc 3D space. Rndle [9] ses ths eqvalence to enable vsalzaton of bent manfolds, wheeas Steane [0] consdes 4

6 ths eqvalence to be a eal opton fo a pesentaton of ealty. Callahan [], beng vey clea abot ths eqvalence, declaes: n physcs we assocate cvate wth stetchng athe than bendng. Afte all, n Geneal Relatvty gavtatonal waves [] ae space waves and the attbton of elastcty to space s ths a mst. The defomaton of space s the change n sze of ts cells. The tems postve defomaton and negatve defomaton, aond a pont n space, ae sed to ndcate that space cells gow o shn, espectvely, fom ths pont otwads. Postve defomaton s eqvalent to postve cvng and negatve defomaton to negatve cvng. Ths s dscssed n Pat..4 Recent Papes on Electc Chage Nonlnea models of electc chage and magnetc moment [3] (05) The engmatc electon [4] (03) Snglaty-fee model of electc chage n physcal vacm: Non-zeo spatal extent and mass geneaton [5] (03) Dalty and patcle democacy [6] (06) Althogh these papes elate to dffeent aspects of o sbect, none pesent a smla dea to os.. Electc Chage. Electc Chage Densty We defne the electc chage densty as: 0 q [q] = () 4 The facto /4π s ntodced fo no othe eason, than to ense esemblance to the Gassan system. 5

7 The chage densty s postve f > 0. The chage densty s negatve f < 0. Necessaly, only two types of electc chage exst, postve and negatve. Let n be the nmbe of space cells n a gven volme V. Snce n = ρ 0 V and also n = ρv we get: V=n/ρ 0 V =n/ρ Hence: (V V)/V = (ρ ρ 0 )/ρ () V >V s dlaton, V <V s contacton.. Electc Chage Q n a Gven Volme We defne Electc Chage n a gven zone of space τ as: Q τ qdτ Electc chage has the dmensons of volme [Q] = L 3 Fo claty, n ths secton alone, we omt the facto /4π n (). Fo the sphecal symmetc case whee dτ = 4π d, fo a gven, the ads of the chage Q, we get the eslt: Q = q4π 0 d = 4π/3 3 ( 0 /ρ). Ths > 0 gves Q > 0 wheeas < 0 gves Q < 0. Fo s, otsde obseves, postve chage n a gven sphecal zone of space wth ads, means moe space cells n the zone than n an n-defomed space (contacton), wheeas negatve chage means less space cells n the zone than n an n-defomed space (dlaton). In Pat, sng Remannan geomety, we elate cvate to ths space defomaton and open the way to the applcaton of GR n sses elated to chage. Note that the eqalty Q + = Q -, of the absolte vales of the bvalent elementay chages means, accodng to the ntegal above, ( 0 /ρ + ) = ( 0 /ρ - ) and hence / 0 = /ρ + + /ρ -. Note that both ρ + and ρ - ae, pobably, fnctons of and not st constants. 6

8 Fg. () sggests smplstc models of postve and negatve chages, both as sphees of ads 0+ and 0-. The contacted space n the sphee wth ads 0+, o Q +, contacts space aond t (ts feld) wheeas the dlated space n the sphee wth ads 0-, o Q -, dlates space aond t (ts feld). In ths model, of chage and ts feld, thee s no physcal sepaaton between the patcle and ts feld, and the ntegal of ρ ove the ente space, fo two bvalent elementay chages togethe, s zeo. Note that n the feld of a postve chage space s also cved postvely. Smlaly, n the feld of a negatve chage, space s cved negatvely. Hence the feld eqaton s non-lnea, as s the feld eqaton of gavtaton. ρ ρ 0 Q(+) 0 ρ 0 Q(-) ρ 0 Fg. () A Chage and ts Feld 3. The Elastc Spatal Vecto and the Electc Feld E By elatng to space as an elastc meda we can se the theoy of elastcty and ts Elastc Dsplacement Vecto =. In Appendx A we show that: 0 (A) Ths, accodng to (): = 4q (3) By defnng the Electc feld vecto E as: 7

9 E = H H = [H] =T - [E] = LT -, (4) eqaton (3) becomes the nown eqaton: E = 4Hq (5) Note that any defomaton (stan) n space s elated to a stess; hence the ntodcton of H. E expesses, theefoe, the tenson n space de to a defomaton n t. Fo a postvely chaged patcle, E ponts otwads and fo a negatvely chaged patcle nwads, as t s n the Maxwell theoy of electostatcs (see Fg. n Appendx A). 4. Colomb s Law Gass s theoem s: d dσ Fo a sphecal sface wth ads we get: Q we get o: 3 dσ 4 and snce d 4 H qd 4Q HQ Colomb s Law (6) E 3 5. The Electc Feld E and Scala Potental Evey vectoal feld can be decomposed nto a feld that s a gadent of a scala potental (the pola pat) and a feld that s a vecto potental (the axal pat), sbect to the bonday condton E 0 at nfnty. Hence: E Α (7) In the smple statc case fo the electc feld: E (8) and, n case of sphecal symmety, n sphecal coodnates: 8

10 E (9) HQ and snce: E we get: HQ [] = L T - (0) Fom: E 4Ηq and E we get 4Hq o: 4Hq Posson s eqaton () In the absence of chage: 0 Laplace eqaton () We can modfy eqaton () to ncopoate the chage densty of the feld, whch s eqvalent to the feld enegy densty. Ths modfcaton tns () nto a non-lnea eqaton that esembles the non-lnea eqaton of gavtaton bt t s ot of the scope of ths pape. 6. The Electostatc Foce Fom (3) = 4q we get q 4, o n tenso notaton q (Appendx B 4 x elates to the metc element g ). Mltplyng both sdes of (3) by gves: q (3) 4 The left-hand tem mltpled by H s denoted by f. f = qh o f = qh (4) At ths stage, f s st a symbol. Afte a few steps, t s dentfed as the electostatc foce densty. Sbstttng the tenso notaton fo, n Eqatons (3) and (4) gves: 9

11 0 x x 4 H x 4 H x 4 H f hence: δ x 4 H f (5) whee s the Konee Delta defned by = fo =, = 0 fo, and 3. Hence f may be egaded as deved fom a tenso: 4 H P. And ndeed: x P 4 H f s dentfed as the foce pe nt volme and P as the stan tenso. If the x-axs s chosen paallel to a lne of foce at any pont, then y = z = 0 and x =, and: zz yy xx 8 H P P P. Ths the pesse pependcla to the sface s eqal to the enegy densty. Fom (4), E = -H, we get the expesson fo the enegy densty n the feld: 8 E (6) Snce f s dentfed as the foce pe nt volme, we can etn to the expesson f = qh, and ecognze the electostatc foce densty: f qe [f] = LT - and the electostatc foce: E F Q [F] = L 4 T - 7. Colomb s Foce Law Eqaton (6) expesses the feld enegy densty of a system of chages. Hence: d H 8 U E whee E s the feld podced by these chages, and the ntegal goes ove all space. Sbstttng E = -, U can be expessed as follows:

12 U 8H τ E dτ 8H Ed 8H Ed Accodng to Gass s theoem, the fst ntegal s eqal to the ntegal of E ove the sface bondng the volme of ntegaton, bt snce the ntegal s taen ove all space and snce the feld s zeo at nfnty, ths ntegal vanshes. Sbstttng (5), E = 4Hq, n the second ntegal, gves the expesson fo the enegy of a system of chages: U q dτ τ Fo a system of pont chages, Q we can eplace the ntegal wth a sm ove the chages U Q whee s the potental of the feld podced by all the chages, at the pont whee the chage Q s located. Fom Colomb s law: HQ whee s the dstance between the chages Q, Q we get: HQ Q U In patcla, the enegy of nteacton of two chages s: HQ Q U and the foce s: F o: U HQ Q Q Q F= H 3 Colomb s foce law (7) 8. Conclsons O defnton of electc chage densty, based on the densty of the elastc space lattce, enables s to elate to an elementay chage not as pont-le and not as a stng, whch ae alen to space, bt as a fnte zone of contacted o dlated space. Necessaly, elementay patcles ae also of fnte sze and have a stcte. Ths ndestandng enables s to deve and calclate the elementay chage/patcles attbtes.

13 Acnowledgments We wold le to than the late Pofesso J. Beensten and Pofesso A. Zgle of the Hebew Unvesty of Jesalem, and Pofesso Y. Slbebeg of the Wezmann Insttte of Scence, fo the caefl eadng and helpfl comments, and M. Roge M. Kaye fo hs lngstc contbton and techncal assstance. Refeences [] A. Ensten: Ideas and Opnons, Wngs Boos NY, On the Method (933) [] R. Pohl, et al The sze of the poton. Nate 466 (7303): 3 6, (00). [3] K.A. Olve et al. (Patcle Data Gop): Revew of Patcle, Physcs Chnese Physcs C 38 (9) (04) [4] A. D. Sahaov: Sovet Physcs-Dolady, Vol. No., P.040 (968) [5] C. W. Msne, J. A. Wheele, K. S. Thone: Gavtaton, P. 0 (970) [6] J. D. Beensten: Phys. Rev. D 7, p. 333 (973) [7] L. Smoln: Thee Roads to Qantm Gavty (00) [8] G. Amelno-Camela axv: asto-ph/00047v 4 Jan (00) [9] W. Rndle: Relatvty Oxfod (004) [0] A.M. Steane: Relatvty made elatvely easy. Oxfod (03) [] J. J. Callahan: The Geomety of Spacetme Spnge (999) [] B. P. Abbott et al.: Obsevaton of gavtatonal waves fom a bnay blac hole mege, Phys. Rev. Let. 6, 060 ( Feb 06) [3] I. Besons and, R. Velande: Fondatons of Physcs Novembe 05, Volme 45, Isse, pp [4] F. Wlcze: Nate Volme: 498,P3 3 (06 Jne 03) [5] V. Dzhnshalev and K. G. Zloshchastev: Cent. E. J. Phys. (03)

14 axv: v5 [hep-th] [6] E. Castellan: Stdes n Hstoy and Phlosophy of Moden Physcs (06) Elseve [7] L. D. Landa and E. M. Lfshtz: Theoy of Elastcty, Pegamon Pess (959) [8] A.F. Palacos: The Small Defomaton Stan Tenso as a Fndamental Metc Tenso Jonal of Hgh Enegy Physcs, Gavtaton and Cosmology, 05,, Appendx A: Contacton and Dlaton of Space, and the Stan Tenso The am of ths appendx s to pove that: 0 Fg. () shows the poston vecto of a spot, p, n space, wth no stan. When stess s appled on space and a defomaton occs, the locaton of p becomes p wth a poston vecto. The vecto s the Elastc Dsplacement Vecto (theoy of Elastcty). The ogn n Fg. () s abtay and does not play any ole n o dscsson. p p Fg. () The Dsplacement Vecto n the Elastc Space In ths Appendx, we show that the dvegence of the elastc dsplacement vecto n an elastc medm eqals the elatve change n the volme dv' dv dv of a staned medm. The followng dscsson s based on a devaton made by Landa and Lfshtz [7]. = ' = x ' x can be denoted by ts components: Let dl' be the defomed dstance between adacent ponts, snce: 3

15 dx' dx d dl dx dl' dx' dx d by the sbsttton of d dx above we get: x dl' dl dxdx dxdxl Snce the smmaton s taen ove both sffxes x x xl and n the second tem on the ght, we get: x x dxdx In the thd tem, we ntechange the sffxes and l. Then dl' taes the fnal fom: dl' dl dxdx whee the stan tenso s defned as: l l (A) x x xx If and the devatves ae small, we can neglect the last tem as beng of the second ode of smallness. Ths, fo small defomatons, the stan tenso s gven by: We see that t s symmetcal: = x x, can be dagonalzed, le any symmetcal tenso, at any gven pont. Ths, at any gven pont, we can choose coodnate axes, the pncpal axes of the tenso, n sch a way that only the dagonal components,, 33 of the 3D tenso ae dffeent fom zeo. These components, the pncpal vales of the stan tenso, ae denoted by (), (), (3). We shold emembe that, f the tenso s dagonalzed at a specfc pont n the body, t s not, n geneal, dagonal at any othe pont. If ths stan tenso s dagonalzed at a gven pont, the element of length nea t becomes: 4

16 dl' δ dx dx dx 3 dx dx 3 We see that the expesson s the sm of thee ndependent tems. Ths means that the stan n any volme element may be egaded as composed of ndependent stans n thee mtally pependcla dectons, namely those of the pncpal axes of the stan tenso. Each of these stans s a smple dlaton, o contacton, n the coespondng decton: the length dx along the fst pncpal axs becomes dx' dx, and smlaly fo the othe two axes. The qantty s conseqently eqal to the elatve extenson (dx' - dx )/dx along the th pncpal axs. The elatve extenson of the elements of length along the pncpal axes of the stan tenso, at a gven pont, s, to wthn hghe-ode qanttes,.e., they ae the pncpal vales of the tenso. Let s consde an nfntesmal volme element dv, and fnd ts volme dv' afte a defomaton. To do so, we tae the pncpal axes of the stan tenso, at the pont consdeed, as the coodnate s axes. Then the elements of length dx, dx, dx 3 along these axes become, afte the defomaton, dx' dx 3 dv' s dx' dx' dx' 3. Ths dv' dv., etc. The volme dv s the podct dx dx dx 3, whle Neglectng hghe-ode tems, we theefoe have dv' dv 3. The sm 3 of the pncpal vales of a tenso s well nown to be nvaant, and s eqal to the sm of the dagonal components = n any coodnate system. Ths: dv' = dv ( + ) o: dv' dv dv o: dv' dv dv and accodng to () we get: 0 (A) 5

17 Appendx B: The Small Defomaton Stan Tenso and the Fndamental Metc Tenso In the pee evewed pape [8] the athos demonstate that the small defomaton stan tenso, see (A), cold be sed as a fndamental metc tenso, nstead of the sal fndamental metc tenso. We qote the Conclson: Thogh the pesent pape, t was possble to demonstate that the small defomaton stan tenso cold be sed as a fndamental metc tenso, nstead of the sal fndamental metc tenso. Also, t was possble to pove that fom that tenso, not only othe mathematcal stctes cold be constcted, bt also anothe fndamental tenso was obtaned; that was to say, we had constcted two of them, μυ, and Bμυσρ. It s thogh these tensos that the gap between pe geomety and physcs s bdged. In patcla, μυ elates the obseved nteval ds to the mathematcal coodnate specfcaton dxμ. Also, the μυ appea as the potentals of the netal feld {6}. Theefoe, t s easonable to assme that, n the pesence of a gavtatonal feld, the μυ s agan the potental whch detemnes the acceleatons of fee bodes; n othe wods, the μυ s the potental of the gavtatonal feld. Ths, a stage has been eached at whch the eslts obtaned can be appled to the theoy of gavtaton {4}. Howeve, that tas that wold not be epeated hee was establshed by Albet Ensten, and fnally fomlated by hm n 96, as pobably the most beatfl of the physcal theoes. Note howeve that space defomaton s a local feate wheeas cvate can also be a global feate. 6