Einstein s Special Relativity Theory and Mach s Principle by Lars Wåhlin Colutron Research, Boulder, CO USA


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1 Einstin s Spcial Rlativity Thory and Mach s Principl by Lars Wåhlin Colutron Rsarch, Bouldr, CO USA Prsntd Fbruary 4, at th AAAS annual Mting in Boston (Gnral Postr Sssion) Abstract: Many historical works on Einstin dscrib his approval of Mach s philosophy and his ffort to incorporat Mach s Principl into his rlativity thoris. Einstin vntually abandond Mach s Principl but with som rsrvations. Howvr, Mach s Principl still prsists and its prsumd incompatibility with Einstin s Rlativity continus to b an obstacl for many in thir attmpts to undrstand Einstin s thory. This ssay intnds to rsolv th issu by showing that Einstin s Spcial Rlativity, in fact, is subjct to Mach s Principl and that th proof can b found in th rlativistic vlocitis of atomic orbits. Evr sinc Einstin publishd his paprs on Spcial Rlativity [,] thr hav bn many scintists who hav not bn fully satisfid with th thory. Prhaps most notworthy is Waltr Ritz, who collaboratd with Einstin in 99, and mor rcntly th lat Profssor Ptr Bckmann, who was th foundr of th journal Galilan Elctrodynamics, th principal aim of which is to rfut Einstin s Spcial Thory of Rlativity. From tim to tim othr scintific journals hav accptd articls critical of Spcial Rlativity, and associations hav bn formd by philosophically mindd groups who do not wholly accpt th concpt of spactim and th rjction of absolut spac and absolut vlocity, as uphld by th Spcial Thory of Rlativity. On such organization is th Natural Philosophy Allianc, which boasts an imprssiv list of mmbrs. Howvr, scintists and philosophrs who bliv that Einstin s Spcial Thory of Rlativity is on of th gratst achivmnts in scinc and irrfutabl, by far outnumbr thos who ar not convincd of its validity. Furthr, th rlativistic vlocity quations hav bn provn rpatdly in high nrgy particl acclrators. I bliv that for many scintists Spcial Rlativity is difficult to undrstand xcpt with rspct to solving th quations. Thr is no doubt that Einstin s nrgyvlocity quation is valid and indisputabl, and on of th gratst achivmnts in scinc. Why thrfor, is thr still scop for dbat? It is th limination of absolut spac and absolut vlocity or, in othr words, th rjction of Mach s Principl [,4], that Einstin himslf was forcd to discard, which crats th conflict. It was
2 in fact Einstin s mathmatics tachr, Hrman Minkowski [5], who introducd th purly mathmatical concpt of spactim, which discardd absolut spac and absolut vlocity, which Einstin rluctantly accptd [6]. It is possibl to vrify that Einstin was corrct in bliving that Mach s Principl should b incorporatd into his rlativity thory. Mach s Principl rquirs that inrtia of mass and consquntly potntial nrgy of inrtial mass must b gnratd by th rst of th Univrs. In mathmatical trms Mach s Principl can b writtn as φ univ = GM univ / R = c whr φ univ is th cosmic gravitational tnsion or th amount of potntial nrgy pr mass gnratd by th Univrs; G is th gravitational constant; c th spd of light; R th absolut distanc to th cntr of mass of th systm and M univ th total mass of mattr within th radius of curvatur R. Tchnically, Mach s Principl can b applid to th Earth and th solar systm by simply using φ sol = GM sol / r = v whr r is th distanc to th cntr of mass of th solar systm and M sol th mass of th solar systm within r. Th gravitational tnsion φ sol at th orbital radii r of th diffrnt plants quals th squar of thir orbital vlocitis. Mach s Principl can b furthr xtndd to our galaxy or to clustrs of galaxis and ultimatly to th Univrs as a whol, at which point φ univ= c. This lads to a vlocity ffct pculiar to Mach s Principl. For xampl, should w want to sling Earth from its orbit at r out to th orbit of Mars at r, thn th amount of nrgy that nds to b addd to Earth is ΔE = m( φ φ ), whr m is th Earth s mass and φ and φ th gravitational tnsion of th solar systm at r and r rspctivly. Th diffrnc in orbital vlocity is thus Δv = φ φ. Howvr, dcrasing th Earth s orbit by th sam amount of nrgy, E = ΔE to a smallr radius r mans a loss of potntial nrgy ( E = loss of nrgy) in th form of friction and radiation or E = m( φ φ ), and th diffrnc in orbital vlocity bcoms v = φ φ. Not, that whn Δ E = E thn Δ v v, which is a consquntial ffct of Mach s Principl. Th Spcial Thory of Rlativity has so far ignord th abov ffct, sinc it considrs mattr at rlativ rst (thus th trm rst mass nrgy E = mc ) and cannot dduct vlocitis from rst or zro vlocity. It can only accuratly b applid to vlocitis that ar producd by an incras in rst mass nrgy or E + ΔE. Einstin s rlativistic vlocity quation can
3 b writtn in a Mach s format as Δv = E E + ΔE φ φ. () univ univ In cass whr nrgy is lost to radiation such as whn lctrons ar capturd in high spd atomic orbits, Einstin s rlativistic quation bcoms obsolt and must b rplacd by a scond quation that can b usd in cass whr loss of rst mass nrgy occurs, such as E E or v = E E E φ univ φ. () univ This bcoms vidnt if w apply both th abov vlocity quations to th innr orbits of atoms and compar th rsults to publishd masurd valus that currntly appar in Th Handbook of Chmistry and Physics (undr Ionization nrgis or Ionization potntials of th Elmnts). Th circumfrnc of th innrmost atomic orbit as dtrmind by Louis d Brogli s wav thory is h v E and solving for E Zq =, ( wavlngth) () 4ε E E by insrting v from Equation () w obtain = E Zq hc ε mn mn + m, (Jouls) (4) whr Z is th atomic numbr;; E th lctron s rst mass nrgy; q th lctron s lctric charg; ε th prmittivity constant and h Max Planck s constant. Th trm, m n /( mn + m ) whr m n and m ar masss of th atomic nuclus and lctron rspctivly, rducs th orbital nrgy to that of th lctron only. For xampl, Z=9 (Cu) yilds a E / q = V or 5.7 V highr than th publishd data. Du to th accuracy of th abov quation a vry small systmatic Compton rd shift of λ = ( cosα ) h /( mc) was rvald in th publishd masurmnts which accounts for th 5.7 V discrpancy at Z=9. Th Compton rd shift could b causd by a rcoil or dflction angl of cosα = ( k log V ) + k affcting th spctromtric masurmnts whr k = and k = ar proportionality constants and th nrgy of th spctra in lctron volts, s Fig..
4 4 Fig.. Compton scattring angl α for th diffrnt lmnts α = cos ( log V ) [ ] Insrting Δ v for Z = 9 from Einstin s Spcial Rlativity Equation () into th abov d Brogli Equation () mn E E Δ =, (Jouls) (5) Zq hc mn + m ( /(ε )) rsults in a critical rror of 74 V highr than publishd data which has promptd invstigators to introduc svral corrction factors such as th DiracFock corrction [7]; slf nrgy corrction [8]; Uhling vacuum polarization corrction [9]; highr ordr vacuum polarization corrction [] and nuclar siz corrction tc., in ordr to match th masurd valus. My prsonal conclusion is that th mathmatics of Einstin s Spcial Thory of Rlativity is only corrct for cass of rlativ incras in rst mass nrgy, as in particl acclrators for xampl, and that Mach s Principl should b includd in th thortical intrprtation of th thory to account for th nrgyvlocity rlationship in cass whr rst mass nrgy is lost, such as in atomic orbits. Th clos agrmnt btwn masurd and thortical valus, s Fig., should prov this point. Th fact that Einstin s Spcial Thory of Rlativity works wll in particl acclrators, but fails for atomic orbits, is quit srious sinc thr ar by far mor atoms in th world than particl acclrators.
5 5 Fig.. Diffrnc in parts pr million btwn masurd valus (corrctd for Compton rd shifts) and th thortical valus from Equation (4). It is rmarkabl that Mach s Principl has to b invokd in ordr to xplain rlativistic atomic orbits whn Mach himslf did not bliv in atoms whil Einstin, on th othr hand, who was first to prov that atoms xist (Brownian movmnt and th photolctric ffct) chos to abandon Mach s Principl Rfrncs: [] A. Einstin, AdP 7, 89, (95). [] A. Einstin, AdP, 67, (96). [] E. Mach, History and Root of th Consrvation of Enrgy, (87). [4] E. Mach, Th Scinc of Mchanics, 6th d., (94). [5] H. Minkowski, Gott. Nachr., 5, (98). [6] A. Pais, Subtl is th Lord, Oxford Univrsity Prss, Oxford, (98). [7[ J.P. Dsclaux, Numrical DiracFock Calculations for Atoms: Rlativistic ffcts in Atoms, Molculs and Solids, d. G.L. Mali, Plnum Prss, Nw York, (98). [8] P.J. Mohr, Procdings of th Workshop on Foundations of th Rlativistic Thory of Atomic Structur, ANL86, (98). [9] E.A. Uhling, Polarization ffcts in Positron Thory, Phys. Rv., 48, 55. [95]. [] G. Källn t al., Fourth Ordr Vacuum Polarization, Kong. Dansk. Vidnsk. Slsk. Mat. Fys. Md., 9,
6 6 WORK SHEET Masurd groundstat nrgis (highst ionization potntial) for onlctron atoms from Handbook of Chmistry and Physics TABLE Z Emnt Enrgy V Z Elmnt Enrgy V Z Elmnt Enrgy V H Na Sc 6.7 H Mg Ti Li.4549 Al 4.4 V B Si Cr B P Mn C S F N Cl Co. 8 O Ar Ni F.76 9 K Cu N Ca Abov valus corrctd for a minor Compton rdshift of λ = ( cos α ) h/ ( m c), whr cos α = ( log V ) , and by using th following quation: q V V = V+ λ ( ). ch TABLE Z Elmnt Enrgy V Z Elmnt Enrgy V Z Elmnt Enrgy V H Na Sc H Mg Ti 668. Li.456 Al 4.5 V B Si Cr B P Mn C S F N Cl Co O Ar Ni F.7 9 K Cu 57. N 6.45 Ca Thortical valus using Equation E Zq mn V q hc = ε n m / ( m + m ) = 8A / ( 8A + ) n n ( m + m ) TABLE Z A Enrgy V Z A Enrgy V Z A Enrgy V H Na Sc H Mg Ti Li Al V B Si Cr B P Mn C S F N Cl Co O Ar Ni F K Cu N Ca h = E4, c = E+8, ε = E, m = E, Protonlctron mass ratio m / = 86, q= E9, E = m = E4 p m c
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