Local Lorentz Transformations and Vectorial Relativity

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1 Uniesidad Cental de Venezela Fom the SeletedWoks of Joge A Fano Noembe, 6 oal oentz Tansfomations and Vetoial Relatiity Joge A Fano, Uniesidad Cental de Venezela Jose G Qinteo Aailable at:

2 Jonal of Vetoial Relatiity JVR (6) - oal oentz Tansfomations and Vetoial Relatiity J G Qinteo D and J A Fano R ABSTRACT: In peios wok it was shown that instead of the known oentz Tansfomations (T) the new obtained Vetoial oentz Tansfomations (VT) wee the tansfomations that tly espeted the postlates of the Piniple of Relatiity and the onsideation of the speed of light as a niesal onstant In this eiew is pesented the way how it is possible to obtain patial onseqenes of the VT, and its appliations to o eal life KEYWORDS: Speial Relatiity, Relatiisti Mass, Relatiisti Enegy and Relatiisti Momentm I OCA ORENTZ TRANSFORMATIONS As it is ommonly expessed in elatiisti liteate, the patial onseqenes of T, nde onditions of simltaneity of eents, o thei oene at the same loation, ae those known as length ontation and time dilation, espetiely In ode to establish a seqene of steps to aie at the oal oentz Tansfomations, let s eall the existent onditions within the analysis of the Vetoial oentz Tansfomations: a) Two inetial systems with elatie moements between them and an obsee plaed in eah system with eqipment to mease time and distanes, peiosly alibated, ae onsideed b) Bease of the elatiity of motion, eah obsee onsides his system as fixed and the othe moing: It is impossible to demonstate whih system is moing, as it was shown by Einstein in 95 ) When both inetial systems oinide a light plse is sent to the spae and measements of its tajetoy ae done by eah obsee within his inetial system withot knowledge of the othe obsee loated in the othe inetial system Ths, eah obsee meases fom his oigin of oodinates a diffeent adio-eto to the point P in the spae, peiosly hosen, whee the light plse aies at d) Eah obsee, in an independent way mst mease that the speed of the light plse is Independent Reseahe, Caaas, Venezela, JonalofVR@hotmailom Noembe, 6 Independent Reseahe, Caaas, Venezela, jafano@yahooom

3 J G Qinteo D and J A Fano R: oal oentz Tansfomations and Vetoial Relatiity Noembe, 6 Unde theses onditions, the manne to obtain a lean expession of the length ontation, fo example of a ba that at this moment is moing at a speed, bt it is known it has a length, meased at est, is by doing simltaneos measements Fo example, et A and B the extemes of the ba moing along the X-axis of fixed system at speed and let x A, x B, x B x A, the measements done by the moing obsee at O, whee the ba is loated So, the fixed obsee will be able to do distane measements on the ba, x A, xb, simltaneosly ( t A t B t ), and he will obtain: xb x A By establishing the VT of distanes and making sitable opeations, follows the efeed expession of length ontation: x B x A ( x B x A ) + ( t B t A ) Fo obtaining the expession of time dilation, let s hae two eents A and B, as fo example two shots of a weapon oing at the same plae ( x B x A, xb x A ) in the fixed system O, at diffeent times ( t A, t B, t B t A t ) Measements done by a moing obsee in fntion of what is meased by the fixed obsee is t A, t B, t B t A t Fo this ase time dilation takes the following lean expession: ( t B t A ) + ( x ) B x A t t B t A t As we hae seen in peios examples, to obsee a lean length ontation it is neessay to hae simltaneity of eents Similaly, to obsee a lean time dilation it needs the eents o at the same plae When simltaneity o same loation ae not met omplex expessions fo length ontation and time dilation aise Obiosly, they ome fom the fat that we ae ompaing measements done fom two distint inetial systems with distint oigins What abot if measements ae done fom the same efeene? Clealy, we ae modifying the oiginal onditions nde VT wee deeloped Obsee that the eqalization of the most of onditions is the sal way to ompae measements: Those onditions left fee of aiation ae peisely those to ompae In this ase, we only want to ompae the measements of a magnitde done by a fixed obsee with those done by anothe moing one, and what is the amont of thei diffeene, if any As a patial eslt, o final goal is to establish if a aiation in the measement of a magnitde is obseed when it is done in moement espet to that at est In this sense we will all the tansfomations loal, bease we ae being allowed measing in o own system of efeene the ale of a magnitde in moement, knowing its est ale Namely, gien this only eason we will all them oal oentz Tansfomations Fo example, when a physiist oneies that a plse of light lasts eight and a half mintes oming fom the Sn to the Eath, and he eeies sh image in his eyes, he knows that Sn is not thee at JVR (6) - Jonal of Vetoial Relatiity

4 J G Qinteo D and J A Fano R: oal oentz Tansfomations and Vetoial Relatiity Noembe, 6 that plae whee he is iewing Sn s image, bt 5 Km fa apat Fom this point of iew, the physiist is thinking in a way that instantaneosly his thoght eflets a the eal sitation of what is in-eal-time is eally oing with Sn s motion In this sense, isal images ae nee eal bt thoghts, like the peios one, depits instantaneosly eality Ths, we will see next that the sitation of simltaneity of eents and oene at the same plae, o the obseed ontation of lengths and time dilation in elatiity an be obtained by eding the onfigation of two distint inetial obsees measing a physial magnitde espet to distint efeenes to the sitation of measing the same physial magnitde fom the same efeene What does this mean? As we know, Vetoial oentz Tansfomations ae the elationships between measements of length and time, done by two diffeent moing obsees inside thei own fame of efeene, withot knowing eah othe When eah obsee meases the speed of light they do thei measements taking as efeene thei own oigin of oodinates, the oigin O in the fist ase, o fo the onsideed fixed obsee, and the oigin O fo the onsideed moing one Despite this, the eslt is that the ale of speed of light obtained by eah obsee is the same Now, let s pose the sitation of measements of physial magnitdes with the diffeent onfigation fo both obsees efeed peiosly, to whih we all oal oentz Tansfomations (T) et s establish that eah obsee knows abot the pesene of the othe obsee, and they agee to mease magnitdes ding the same peiod of time, by taking as efeene the same oigin of oodinates, in ode to ompae thei eslts These measements, of ose will yield diffeent eslts if ompaed with those obtained thogh Vetoial oentz Tansfomations, whih ae done by taking diffeent oigins of oodinates We will see also that this new onfigation will gie s patial elationships that ae not dependent on the oientation of the body s moement Remembe the diffeent tansfomations of magnitdes sally handled in the Speial Theoy of Relatiity: ongitdinal o Tansese oentz Tansfomations In ode to systematize the ideas and apply them fo taking hold of a eal ompaison of measements let s establish, fo obtaining the efeed oal oentz Tansfomations, the following two onentions: ) Fom now on, both obsees will do thei measements by taking the same point of efeene et the oigin of the fixed obsee be this efeene Fo example, if the fixed obsee O meases the adio-eto of a pojetile sent to the spae, see Fig, the obsee on the moing system O will mease a simila adio-eto R fom this same efeene of the fixed obsee, sh that R will flfill R R, with the definitions of the aiables R and R gien below: R R ; fo R () t + t + t Then, the measement of moing obsee is elated to that of the fixed one, only by the saling fato It is impotant to notie that T ae also ompatible with Maxwell eqations JVR (6) - Jonal of Vetoial Relatiity 4

5 J G Qinteo D and J A Fano R: oal oentz Tansfomations and Vetoial Relatiity Noembe, 6 P R R + > R R R (/) t (/) t (/) R R R + t t + (/) (/) θ t R R (/) os θ Fig ) The moing obsee O will not send any plse of light (o pojetile) Ths, he will mease a nll displaement of pojetile So, the adio-eto of his moing system,, will be the only measement ompleted Theefoe, fom the geneal VT expessions peiosly obtained and the onentions applied, the oal oentz Tansfomations (T) ae obtained as: t t t ; t t t ; R () In sm, the moing and fixed obsees ae efeing thei measements, R, t, t, and, of the moing oigin O espet to the same point of efeene: the oigin of oodinates of the system O Gien that time is a eto with spatial omponents, it an be thoght as efeed to the spatial oigin O As we obsee in (), time and distane etos ae elated by a haateisti-saling fato The saling fato, with a ale less than nity, in the ase of time is a mltiplie within eah omponent Fo distanes, it is a diide In othe wods, if any omponent of a paamete has the same type of fato the ontation o expansion is the same in any dietion, fo instane, a sphee shold expands nifomly in all dietions aoding to the saling fato depending on its eloity and the speed of light JVR (6) - Jonal of Vetoial Relatiity 5

6 J G Qinteo D and J A Fano R: oal oentz Tansfomations and Vetoial Relatiity Noembe, 6 Howee, it is impotant to point ot that the T ae efeed to measements elatie to the same point of efeene: oigin O This is ompletely diffeent to what is done fo VT, whee eah obsee does measements elatie to its own efeene system Namely, the tansfomations efeed to T ae diffeent to those of VT With those two onentions in mind, we will not be woied abot loation oinidene o simltaneity of eents The elations () imply that in T eah physial magnitde obseed by an fixed obsee, by ite of its dependeny on eloity, in a te way eithe ontat, expand, gowth o ede, with the same saling fato in all dimensions, independently of thei image o how we see them Fo example, if on the moing system an obsee at O meases a ba of length, lasting a time t in his measing, the obsee on fixed system at O will mease this length as and time t as t The position of the ba in system O is not eleant; what is impotant is that it is at est fo the obsee at O and moing on elatie to O Ths, the elationship between both measements, aoding to T, will be: ; t t ; t () t This indiates that an obsee in a stationay system O meases onto a moing ba at eloity, a ontation fom its oiginal length to, no matte whih is the position of the ba in the system t O, and a time dilation fom to t, as it is shown in eqations () Anothe eleant haateisti is the following one: oentz fatos in T at as saling fatos between measements done at O and at O, fo any magnitde, no matte if this is a diffeential magnitde o an integal one In othe wods, oentz fatos ae simply saling fatos between sh measements What is the eal meaning of T expessed in eqations ()? Fist of all, eah omponent is affeted by the oentz fato in the same way, namely, ontating lengths Fo example, If instead of a ba the obsee in the moing system had had a sqaed ba, whose aea, as we know, is the podt of two lengths, then the obtained T of sh aea fo an obsee at the fixed system, wold be theefoe the podt of two ontated lengths (late we will se this T haateisti of aeas): S S S S S S (4) A olme V, meased fom O is elated to the olme meased fom O, V, by the haateisti podt of the thee ontated lengths gien below in (5): JVR (6) - Jonal of Vetoial Relatiity 6

7 J G Qinteo D and J A Fano R: oal oentz Tansfomations and Vetoial Relatiity Noembe, 6 V V V V V V (5) The eloity of the oigin O is obtained by diffeentiating the displaement of O espet to time and sbstitting known Ts The T fo eloity beomes: d d d R (6) At this moment we ealize that eloity of the moing system O, plays two oles: eithe as a sala,, when it is inside the saling fato, in whee both obsees see eah othe moing elatie to themseles in the same line nde onditions of VT O, as a eto, meased by the obsee at O into his own fame by taking as efeene the oigin of the othe system O, nde onentions of T It is impotant to be onsios with these two diffeent onepts! Afte doing this neessay paenthesis, let s ontine: T fo aeleation is obtained in the same manne as in (6): d d d a a a (7) It is also neessay to say at this moment that oigin O old also hae a motion along an inetial ilinea path following an inetial moement with aiable eloity Fo example Eath has an ndobted inetial ilinea moement aond the Sn, and althogh it aeleates going to peihelion and ede its speed afte peihelion going to aphelion, we don t feel anything, bildings maintain thei etiality, eqilibim of any kind is peseed, et So, with the fond tansfomations in eqations (6) and (7), we wold expet to obtain also T in Dynamis et s do some emaks As we will demonstate next, it is possible to apply the Vetoial oentz Tansfomations (VT), to an inetial system of oodinates with ilinea moement, with espet to a fixed system, loated in a point thoghot the ilinea tajetoy of the moing system JVR (6) - Jonal of Vetoial Relatiity 7

8 J G Qinteo D and J A Fano R: oal oentz Tansfomations and Vetoial Relatiity Noembe, 6 P O t EARTH t O β θ SUN R Fig γ P osγ O β O sinγ dx osγ dx sinγ osθ θ X X R Fig JVR (6) - Jonal of Vetoial Relatiity 8

9 J G Qinteo D and J A Fano R: oal oentz Tansfomations and Vetoial Relatiity Noembe, 6 We an establish that, inetial systems ae not only those with nll aeleation, bt those whee the sm of ating foes is nll These inlde not only those with nll aeleation in etilinea moement, bt also those in ilinea moement with a onstant Angla Momentm Fo the moement of Eath aond the Sn, the smmation of the gaitational foe of Sn onto Eath pls the Eath s entifgal foe gies a nll eslt, eason why the Eath moement is inetial aoding to sine we hae defined it In this way, eath s moement is neithe impeded no eased by any additional extenal foe We will ty to epode this moement in Fig, whee the fist obsee is on the moing system, Eath, at O, and the seond obsee will be fixed on the ellipti path at the neaest point to the Sn, the peihelion R et s denote, as the distane between Sn And Eath at the moment when obsees stat measing the moement, and, the genei position of Eath By taking a lose iew at the ey beginning of measements onto this moement, fo two dimensions, see Fig Say, when O and O oinide, a plse of light is sent foming an angle β with X axis, see Fig, and an angle γ between the tangential eloity of O with Y axis as it is shown in Fig et s eqally defineθ, as the angle swept by adis fom R, to the new position of the moing obsee afte a peiod of time At this moment light plse has eahed point P Fom Fig and, we an establish the following elationships: dx k( dx sin γ ) dy k( dy osγ ) (8) Fom the same gaphs, we an establish that: sin γ d( R osθ ) osγ d( sinθ ) (9) Gien that the light speed is the same meased by any obsee, it mst flfill: dx + dy dx + dy () Sbstitting dx, dy, by thei expessions (8) and (9) into (), simila expessions peiosly obtained fo etilinea moement ae ahieed: dx x dx sin γ dy osγ dy sin γ sin γ x x + osγ y y sin γ x sin γ y x + osγ osγ + osγ y y ( JVR (6) - Jonal of Vetoial Relatiity 9

10 J G Qinteo D and J A Fano R: oal oentz Tansfomations and Vetoial Relatiity Noembe, 6 Namely, d d ; d d () These eslts show that the stte of diffeential VT fo ilinea is the same peiosly iewed fo etilinea moement Obiosly, all this indiates that appliation of T to ilinea moement is also alid, whih will allow s to ontine deeloping Relatiity within only one theoy The peios analysis was pesented fo the patila ase at the beginning of measements to diffeentiate the lengths of the diffeentials dx and dx The same sitation an be displayed fo the genei point P that the plse of light is dawing in the spae, whee the same elationships ae Also alid (see Fig 4) Namely: x sin γ dx sin γ x dx y osγ dy osγ y dy dy dy P y O x dx γ X y sinθ osγ O β x sinγ osθ θ dx X R Fig 4 JVR (6) - Jonal of Vetoial Relatiity

11 J G Qinteo D and J A Fano R: oal oentz Tansfomations and Vetoial Relatiity Noembe, 6 et s ontine obtaining othe dynami tansfomations, fo instane, that fo angle between inetial systems This magnitde emanate fom the elation between ilinea length of a s and length of adis R Bease both magnitdes ae lengths, oentz fatos anel ot, and angle beomes inaiant to T (this eslt is diffeent to that of explained by Einstein in SRT []): s ds α s R R s R α α; dα ds R R ds R dα dα () In this way, angla eloity tansfoms as: dα dα dα ω ω ω (4) In next Reiew we will obtain the T of Foe and othe physial magnitdes V CONCUSION We hae obseed that oal oentz Tansfomations (T) gie s the te dynamial ale of a physial magnitde whose est ale is known, namely T infom s abot the eal dependene a physial magnitde has on the speed of light and on its own speed in spae In this way, the Theoy of Relatiity stops being a mysteios and omplex sbjet, ndestood only by a few indiidals, to beome something simple and easonable, familia to anyone, and eealing to s a new and simple physial law that goens the moement of the bodies in spae REFERENCES [] A Einstein Z Elektodynamik bewegte Köpe, Annalen de Physi 7:89, 95 English esion pepaed by John Walke On the Eletodynamis of Moing Bodies [] A Einstein Ist die Tägheit eines Köpeson seinem Enegiegehalt abhängig?, Annalen de Physi 8:69, 95 English esion pepaed by John Walke Does the Inetia of a Body Depend pon its Enegy ontent [] P Gibbs, J Ca; D Koks Does mass hange with eloity 997 [4] P M Bown Physis Wold 5 Speial Relatiity Setion ongitdinal and Tansese Mass JVR (6) - Jonal of Vetoial Relatiity

12 J G Qinteo D and J A Fano R: oal oentz Tansfomations and Vetoial Relatiity Noembe, 6 [5] A Aliotta, G Amellini, P Caldiola, B Finzi, G Polani, F Seei, P Staneo and M Pantaleo Cinqant anni di REATIVITÁ Pefazioni di Albet Einstein Seonda edizione -No-955 Editie Uniesitaia Fienze Italy Page 97 [6] C G Adle, Does mass eally depend on eloity, dad, Am J Phys, 55(8) Agst 987 [7] Okn, The Conept of Mass, Physis Today, Jne 989 [8] M Sahs, On the Meaning of E m, Int J Theo Phys, Vol 8(5) (97) [9] J A Fano R, Vetoial oentz Tansfomations 6 EJTP 9 (6) 5-64 [] R Gatea, W Sain Físia Modena Segnda Ediión MGRAW-HI, Méxio, Page 54 Tanslated fom the seond English edition of SCHAUM S OUTINE OF THEORY AND PROBEMS OF MODERN PHYSICS, 999 [] P A M Dia, Piniples of Qantm Mehanis, 4th edition (Claendon, 98) [] M Sahs Einstein ess Boh The Contining Contoesies in Physis Open Cot Pblishing Company, a Salle, Illinois, 988 Page 9 [] M Alonso and E J Finn Físia Vol III FUNDAMENTOS CUANTICOS Y ESTADISTICOS Vesión en español de: C A Heas y J A Baeto Aajo Editoial Fondo Edatio Inteameiano, S A (97), página 6 JVR (6) - Jonal of Vetoial Relatiity